I am an US citizen living abroad, I understand that I need to pay taxes even abroad, but how do I pay taxes if I'm not working for a US company?
You still need to file. Fill out the PDF form from the IRS web site and mail it to them.
If you're working for an overseas company you can claim the Foreign Earned Income exclusion, which is about US$85,000 per year. If you earn less than that you will pay no tax on your overseas income to the IRS.
If you're self-employed, you will still need to pay the employers contribution to Social Security and Medicare, as these are not covered by the exclusion. I found this situation too hard to deal with on my own, and found a local accountant who specialises in US tax matters.