Hello Cole, thanks for the question. I'm sorry for your loss.
I can tell you generally where to start.
Firstly, talk to other relatives or children of your father, see if anybody else is interested in helping to wrap up the estate. None of you has to cover his bills, so don't worry about that. But maybe he has close friends or other family who would like to help, as a way of saying goodbye perhaps.
If there's nobody other than you, then you need to bring an action to the local probate or superior court to get an order that names you as the estate administrator/executor. The note from your father isn't exactly binding or "legal", but it's evidence of what your father wanted, and if nobody else wants to do the job then the court will make an order that you can do it. If you want, have an estate lawyer help you with this paperwork.
Once you have that order, then there's a list of stuff to get done. I assume that funeral arrangements have already been taken care of. Make a list of assets and debts, give copies of the death certificate and your court order to the banks which your father dealt with, set up an "estate account" in your name where you'll put his monies, inform his debtors, do the terminal tax return, look for insurance policies which he might have had and go from there.
It's an annoying process and can take some time, but someone has to do it, and your father wanted it to be you.
Does that help you get started? If there's more to discuss then please reply. If I've answered you then I'd appreciate a positive service rating please.