I am in the same situation as i am a Canadian living in virginia and want to get one. What you are first going to have to do is figure out if you are a non-immigrant or not. Basicly are you allowed to permanatly live in the US or not. If you are eg have a green card, you can buy pretty much like a citizen would but they will need to get information from your green card. If you are not allowed to permanatly live in the US then you cannot own firearms or ammunition execpt for a few specific cercumstances, the first being that you get a waiver from the DOJ, the next being that you have a hunting license/waiver for a US state, and lastly if you are a official rep, official or distinguished visitor of a foreign goverment that has been accredited by the US goverment/state department.
basicly for most of us it boils down to either get a hunting license or get a waiver from the DOJ. Now i dont know about the local or state laws for where you live, that is something that you will have to look into. These are just the federal rules. Also when they do the criminal check they will check with the INS to make sure that you are in the country legaly, so that could make it take a bit longer.
hope this helps you out some
pyro_