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.
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.
You really will struggle to complete this task without the proper tools, I can't actually imagine how you would do it successfully. Criggie's answer helpfully details some good ways to go about borrowing the tools.
As people have suggested in the comments, if you are changing to an 11-speed cassette, you need to change the shifter(s) and derailleur(s) as well, but even then the cassette might not even fit on your old wheel - it most likely won't fit, meaning you need a new wheel too. Affording all of this but not being able to afford the basic bike tools required, even cheap versions, doesn't sound quite right.
My answer is, if you can afford to upgrade to 11 speed, you can afford to buy the tools, or to pay the labour, or to travel and visit someone who will lend you the tools. If you can afford something in between that and nothing, you would be better served by replacing the cassette (and chain) with new versions of the existing items. A well maintained 7 speed bike will be far better for you in several months time, than a poorly maintained 11 speed bike.
I once used ISIS spindle to undo cassette lockring. Even though the shape and diameter is similar, there are 12 splines on lockring opposed to 10 on ISIS spindle. There will be some contact, that might be sufficent. Just undo crank on one side and use crank on the other side as lever. However this probably won't work for properly torqued lockring.