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it's very simple. I believe you just go to the FBI site. You will need to locate where in your town you can get fingerprinted. Then you just fill out the form, which I believe you can do online and send the fingerprints to the FBI. The turn around is very quick. There are fees, of course, both for the background check and the fingerprinting.

Much appreciated.

To be clear, I’m not in  the United States right now, I am in south Portugal. Do you mean I can get fingerprinted in a town in Portugal? And that will be valid to send through the form on the fbi site?

I am an US Citizen going through this process right now (between step 2 and 3).

If you are an US citizen unmarried with an US passport, then legally you only have 90 days to be in Portugal.   If you have been here longer then you will probably get deported next time you use your passport,  As to getting the FBI background check, just follow what the above poster described, but it will have to be done in the US.  As to your question about insurance, yes, in order to get residency, you will need to show proof of insurance that will cover all expenses in Portugal or monetary means to be able to pay for everything.

Normal steps to get  residency
1.  US Passport - 90 days stay
2.  Temporary Residency - 4 month stay (is extended once applying for residency)
3.  Resident - first for 1 year next for  2 years