You have to go to court with documents that the person is incapacitated from at least 2 medical professionals. The cost is paid by you up front and can run from $3000-$8000 depending on the state and whether the person fights it. You really ought to consult a lawyer because you don't just "get paperwork to be a guardian".
Where I live, there are people who help with guardianship for free at the Probate Courthouse. This is in a large city. There may be similar resources in your nearest large city. You can ask them about the paperwork needed, how to fill it out, there may be a legal aid phone line, and you can go that route if you're not in financial position to pay heavy attorney fees. Just some additional info since we're on the topic... :) Also, if you're handy with searching online, or know someone who is, they could find out where those types of resources could be found.
If you have the means, a lawyer is definitely the way to go to handle a guardianship case for you. You will need to establish in a court of law that the person is mentally incompetent. Once that is done, you can apply for guardianship. Any other kin will be notified and will/should have a chance to contest your appointment as guardian, should they want to do that. I mention these things because -- like others here said already, guardianship isn't something that you simply request and it's granted. There is a process. If your Loved One truly needs a guardian, there is also pro bono help available, in my experience.