The Hebrew literally says, "God, your God". This translation is straight-forward and, as far as I can tell, undisputed. The interesting question concerns the interpretation:
Ps 45 seems to be a psalm about a king, perhaps used in the ordination of kings:
Psa. 45:5 Your arrows are sharp in the heart of the King’s enemies;
The peoples fall under You.
Psa. 45:9 Kings’ daughters are among Your honorable women;
Psa. 45:10-11 Listen, O daughter,
Consider and incline your ear;
Forget your own people also, and your father’s house;
So the King will greatly desire your beauty;
Because He is your Lord, worship Him.
Either way, the king is the main character of the psalm. The "sons of Korah" who wrote the psalm (according to its inscription), were music leaders in the temple (1Chr 6:31-37), which was in Judah. This is therefore about a king or kings of Judah, of the line of David - kings who the Old Testament says were (figuratively) anointed and blessed by God.
This anointing by God is what v7 is speaking of:
Psa. 45:7 You love righteousness and hate wickedness;
Therefore God, Your God, has anointed You
With the oil of gladness more than Your companions.
God is called "God, your God" to distinguish him from the "god" of v6, which is referring to the king himself in exalted terms. The book of Hebrews refers to this as a Messianic prophecy, which makes sense in light of the Messiah being the "son of David", the "branch of the root of Jesse", the one prophesied to restore and fulfill Davidic kingship. Those who see this as exclusively being a messianic prophecy, compare it to Ps 110:1, "The Lord said to my lord, sit at my right hand until I make your enemies your footstool," arguing that the reference to God's God in Ps 45:6-7 cannot be resolved except by Jesus. I respect this opinion, but disagree: calling a king "a god" is no more blasphemy than calling an excellent meal "divine" - it is simply an term of great respect and appreciation. Neither do we object to the term "godly" when describing a man of good moral fibre. Likewise, no one saying of a celebrity, "he/she is a god" means that the celebrity is a non-human deity with supernatural powers. Calling the Davidic king "god" in no way elevates him to divine status - he is clearly distinguished from God by v7, which says that the king was chosen and anointed by God, who is the king's God.
That being said, I am not saying that the text in addition to primarily functioning as a psalm about a king or kings does not also double as a messianic prophecy.