Is there any way to remove a rear cassette WITHOUT a cassette wrench? I've tried to and tried, but just can't do it. Can anyone help me with a way to remove my crappy old 7 speed cassette and install a brand new 11 speed without using a cassette wrench? I'm not rich enough to own one.
Use the correct tool.  There really is no other good answer.

Bodges that could get the old one off, maybe


a hammer and cold chisel - likely to upset the threads on the freehub.
a grinder and cutoff disk - guaranteed to damage the freehub, plus throws sparks and metal shavings into the mechanism.   I've actually done this to a cheap bike where all I needed to save were the spokes, and not the freewheel or hub.  Most enjoyable.
a gas axe/torch will definitely cut the lockring off, but you'll also damage the hub and spokes, and possibly the rim too.


Practical Alternatives

You need a cassette lockring removal tool.  As per comments they're a few dollars from China, or about 10x that from your local bike shop immediately.

 

Park FR-5, likely to last decades.

You will also need some way to hold the old cassette to undo the lockring.  The correct tool is a Chain Whip

 

Park SR-1, again expect 30 years service out of this tool.  It costs three times as much as the locknut tool.  I have managed to bodge this by using a length of worn old chain and a pair of vise-grips.



What can I do?

There exist Bicycle Co-operatives who exist to help you with your bike.  They're generally free or donation-if-you-can levels.  And they will have these basic tools for you to use, and may even run workshops.  For me locally, RAD (or Recycle-a-Dunger) is a good service, and can be found at http://www.radbikes.co.nz/

Another option is any kind of "mens shed" which are global, and tend to be populated by some very handy chaps.  Example  http://www.kinrossmensshed.org/what-we-do.html

A third option is to ask around anyone you know who cycles.  Worst that can happen is they say no.

Lastly, you might be able to take your (clean) wheel into your Local Bike Shop and ask for help.  If you're a customer, a LBS might be able to help by quickly slapping it through, for minimal or even no charge.