What's the proper technique for starting uphill? I am a beginner cyclist, having not learned how ride a bike as a kid, I've started riding from scratch 3 months ago. Local "bike buddies" have helped a lot.

I'd like to ride my bike to work, which is only 2km (1.24 miles) away from home. However, the entire path is a climb - max grade 4% for about 2 blocks, which is tolerable. While I can ride uphill, I can't start uphill. When I try, my front wheel tilts sideways and either by reflex I put a foot down, or if I am able to balance, I start riding diagonally, but bikelanes are too narrow for me to do that.

I ride a freewheel/singlespeed, so there are no gears for me to shift. Also, I haven't learned yet how to ride standing, so that is not an option for now.

Am I missing something in the starting technique? The only thing I could think of is that maybe I'm not pushing the pedal hard enough when starting, but I feel like pushing harder is what makes the front wheel tilt, due to pressure on handlebars.

I've read the Sheldon Brown article, and I am able to apply the proper starting technique he described when in a flat or downhill terrain, and I think it's because the bike gets enough speed only with my weight on the pedal.

I've also read How to start uphill?, and it's a different scenario, it was informative but didn't help me.

EDIT: I (finally) bought a floor pump and realized that part of my problems were uncalibrated (underinflated) tires. It doesn't make it easy to start uphill, but it is less harder. I think it'd be so because the first stroke gives the bike more momentum.
Based on the Sheldon Brown article here's my approach. I find that putting the pedal 45 degrees forward of top dead centre is too far for maximum starting impulse - I go more vertical than that. I also start with my strong foot resting on the pedal, but that's partly because that means I lean the bike away from traffic and towards my single pannier when stopped. 

Starting out on a hill I'm forward of the saddle. I push forward with the foot on the ground as I step up on the pedal, which puts the saddle under me. 

But this depends on being in a suitable gear. And as you don't have a choice in this respect that sets an upper limit for your starting gradient. Unless your gearing is too high for the riding you do (and don't rule this out) you should be ok starting out on 4%. I have about a 6% start regularly, in a bike lane, and would be in something like 38x13 on 700C wheels - so a similar ratio to many single speed commuter setups I've read about. A little lower would be better, a little higher is possible. 

One tip if it's a traffic light start in a bike lane: start as  wide as you can in the lane and get going as soon as sensible (this is both a personal and a local matter). That way any wobble out of the bike lane is done while you've got some space around you - but you've got room to wobble towards the kerb.