Can I get a green card if I came to US under fake name?

Entering the United States on a fraudulent document is typically a felony and then depending on the situation there can be other associated crimes.  Assuming your fingerprints are into the US Government system and they are associated with the identity you used to enter the country, if you apply for permanent residency using your legal name, you will automatically be identified as a criminal who committed document fraud, which typically results in jail time in the US, followed by deportation, and permanent bar to enter the US.  If you ever used a false passport, green card or visa to legally cross the border or enter through an airport, you have already broken the law, and it is best not to apply for adjustment of status.  While Obama Amnesty is going to be very lenient on illegal aliens, it is highly unlikely that document fraud will be forgiven.