Q:

How to filter outside air during wildfire to safe breathable level?

I've had an experience of having outside air heavily polluted by wildfire smoke, and now I'm reading up on the wildfire happening in Southeast Asia. When the air outside got too polluted to breathe, I covered a plain old fan with a wet cloth and closed all windows. That helped me filter out the air somewhat, but I would like to know if there are more effective measures to take in such situations.
The CDC's site says you should use air conditioner's air filtering function if you can't relocate. Let's say you can't, what can you do to make the air in your home breathable?

A:

The short answer is that if the air in or around your home is not safe and breathable, you need to leave the area and possibly seek medical attention. You're extremely unlikely to be able to MacGyver your way out of a truly life-threatening air quality event caused by a wildfire (or any comparable industrial accident). Follow the advice laid out in the CDC article you linked. There is no One Weird Trick that Firefighters Hate method for avoiding relocation/evacuation.
If you belong to a sensitive group—elderly, asthmatic, pregnant, etc.—there may be shelter space available at a local hospital or other public, professionally-ventilated building during extreme air quality events. Watch the news, listen to the radio, contact your local government office, etc. to find out about this sort of resource.
Understanding short-term ambient smoke exposure
It's reasonable to be concerned about exposure to smoke at levels that are irritating or unpleasant, even if not immediately life-threatening. What most people want to know is, what are the long-term harms of breathing this smoke? How can I stay comfortable in my home while also protecting my health?
The California Air Resources Board's wildfire smoke guide for public officials summarizes the issue this way (emphasis mine):

One concern that may be raised by members of the general public is
  whether they run an increased risk of cancer or of other chronic
  health conditions (e.g. heart disease) from short-term exposure to
  wildfire smoke. People exposed to toxic air pollutants at sufficient
  concentrations and durations may have slightly increased risks of
  cancer or of experiencing other chronic health problems. However, in
  general, the long-term risks from short-term smoke exposures are quite
  low. Short-term elevated exposures to wildfire carcinogens are also
  small relative to total lifetime exposures to carcinogens in diesel
  exhaust and other combustion sources. Epidemiological studies have
  shown that urban firefighters exposed to smoke over an entire working
  lifetime have about a threefold increased risk of developing lung
  cancer (Hansen 1990). This provides some perspective on the magnitude
  of potential risks from short-term wildfire events.

"Short-term" in the context of a wildfire is usually on the order of a few days. Though the fires themselves can burn for much longer—one recent wildfire in California burned for over a year—wind and weather patterns blow the smoke in different directions at different times, so that it's very rare for any particular area to be smoked in for more than a few weeks, unless it's surrounded by or right next to the fire (in which case I would expect the area to have been officially evacuated).
Being smoked in can be very unpleasant (I know from experience) but unless you have a pre-existing respiratory condition it's unlikely to pose a real danger to the average adult. When the smoke gets so thick that it poses an acute risk to the average person, your only reliable solutions are to leave the area or find shelter in a building that has a well-engineered ventilation system. Local agencies may direct sensitive groups to official shelter areas, often shopping malls, as their systems are already designed to supply comfortable, conditioned air to a great many people. Even if not officially designated as a shelter, an large, air-conditioned commercial building (not an apartment building) is almost always going to be better-equipped for this purpose than your home.
Keeping smoke out of your home
If you're very concerned about your exposure to smoke and your ability to relocate during extreme air quality events, hire a licensed and bonded HVAC contractor who specializes in high-efficiency central heating and air systems.  You need a tight building envelope and a positive pressure ventilation (PPV) system if your goal is to prevent smoke infiltration, plus better-than-consumer-grade filtration to effectively remove the particulates that do make it into the home (e.g., on your clothing).
This will be an expensive retrofit, generally more so the older the home is (newer homes are built with generally tighter envelopes). I'm not sure PPV is even feasible in the typical single-family home and, if it is, it will cost an arm and a leg—I mentioned it half as a joke (think space suits and airlocks). Realistically, a professional whole-house job is probably not an option, and you might be thinking—well, I'll just do it myself on the cheap. Not so fast.
Olin's answer suggests that sealing your house well enough to make a DIY fan/filter arrangement workable is "hard to do" and while that is correct, it's also a bit of an understatement. You could do more harm than good by blowing in outside air that's not filtered properly—or, for that matter, by backdrafting the exhaust from a gas appliance into your home. Improperly ventilated gas appliances do kill people every year in the U.S.; while it's statistically less likely than being hit by lightning, a bad DIY job in a house with no CO detector is a great way to increase your odds of winning this fabulous prize.
So what you want to do, if this will be a DIY project, is focus on simply sealing your building envelope—check and replace weatherstripping and door sweeps, feel for drafts around windows, doors and vents. You'll still want to hire an HVAC company to come do a home energy audit/weatherization, but this will be relatively inexpensive. They'll check floor registers, bathroom fans, range hoods and many other potential draft sources and seal any they find as best they can—few homeowners have the experience and equipment required to do a competent job at this themselves. The more you prevent infiltration of smoke while your doors and windows are closed, the less demand you place on any consumer-grade indoor filter you buy.
Then, go out and buy HEPA filtration units sufficient to cover your living area. Specifically, you're looking for a device that sits on the floor or on a shelf that recirculates the air in the room through a removable filter. Don't be fooled by "artificial cheese flavor product" style marketing tricks; many cheaper units will say "HEPA" on the box and make claims that sound impressive but will not actually be tested and proven to meet this (or any) specific standard:

HEPA filters, as defined by the DOE standard adopted by most American
  industries, remove at least 99.97% of airborne particles 0.3
  micrometers (µm) in diameter.

The diameter part is important; not all particle pollution is created equal and there are any number of ways to filter large particles. What would you think if I sold you a filter that was guaranteed to remove 99.97% of airborne particles 100 µm in diameter? You might not realize I'm selling you a cotton shirt, and the particles it removes are the size of sand on a beach. Look for products that have been tested and certified to deliver real HEPA filtration (PDF). They cost more, because testing isn't free—but cheaper models may or may not protect you at all from the smaller particles that we believe cause the most harm.
Refer to the manufacturer's specifications to see how much square footage the unit covers and keep in mind that closed doors, small hallways and stairways all tend to inhibit airflow, so a smaller filter in each living space is often more effective than a larger filter in a central area. Follow the recommended filter replacement schedule.
Avoid "ionic" type units, particularly the ones in this list; some of them at least won't poison you but I'm not confident they offer any real filtration benefit. In an extreme situation where you've exhausted all other options and can't leave the home, I guess bring all the filter units you have into one room, shut the door and turn them on—but I make no guarantees it will protect your health, especially if you fail to follow the safety recommendations of organizations like the CDC. (At the prepper level of paranoia, you could go shopping for an expensive respirator with a dedicated air supply but this seems a bit like bringing your own personal parachute on an airplane—if you expect to need it, you probably shouldn't make the trip at all.)
If you have central air, there should also be one or more filters on your return air ducting. These are sometimes only designed to trap larger particles but you should learn about your system and the filters it accepts, and perform regular maintenance to keep them in working order. If you heat your home with a fireplace or an older wood stove or you are a smoker, you have much more pressing indoor air quality concerns than ambient wildfire smoke; deal with those first.

A:

There are are three independent problems with air containing "smoke" from a fire:
Particles
Poisonous or unhealthy combustion gasses.
Oxygen depletion.

Unless you're really close to the fire, #1 is the biggest problem.  As you already found, plain old mechanical filtration is the simplest way to address this.  There are commercial air filters for HVAC systems, but those are usually more designed to filter out dust, not the much smaller smoke particles.  There are filters for that too.  Just about any pleated paper filter will likely be good enough.  Examples of common ones include the air filter for the engine in your car, and the cabin air filter.
The best answer is therefore to go get commercial filters of the right kind and force air thru them somehow.  The "somehow" is the difficult part.  Ordinary window fans and the like are optimized for moving large volumes of air around, but generally can't handle much pressure drop.  The type of filter that will remove smoke particles will cause enough of a pressure drop that a typical "room fan" would probably move very little air thru the filter.
A better alternative, keeping to reasonably available parts, is to use a "exhaust fan".  These are meant to move air thru a pipe a ways, and are meant to handle a little back pressure.  You would have to rig up your own jig to have such a fan suck in outside air thru a filter, then exhaust it into the house.  The air flow will be low, so you have to be careful to keep the house otherwise shut as tightly as possible.  You want the only air input to the house to come thru the filter.  It will then find its way out thru the inevitable cracks on its own.  You want the flow to be enough so that air only exits thru these cracks, even when the wind is blowing.  That will be hard to do with most houses.
A single exhaust fan, even if the filter doesn't decrease its air flow much, won't likely be enough for a whole house.  If you have to rig your own filter, it gets even worse.  Paper towels over a window screen will probably work well enough to filter smoke particles, but the performance of such a filter will be poor compared to commercial pleated paper filters designed for the purpose.  You will need much more filter area to compensate for the increased back pressure.  

