When you file form I-485 you will want to explain the discrepancy and fully document the name change. That will be easier if your name was changed legally, or if you have corrected your birth certificate from the original name to the name you use, and you can provide authenticated records (with translation, if necessary) to establish the name change.
As you don't seem to have changed your name through legal process, you should also explain why your nation didn't verify your legal name before issuing a passport with the name you use. Changing your name now won't explain why your passport was issued in the 'wrong name' prior to your name change petition.
It would be a good idea to have an immigration lawyer assist you with the process, to avoid delays and other problems.

