This psalm describes a royal wedding. 

The king is mentioned in 45:1, 5, 9, 11, 13, 14, 15.

45:1 The writer speaks of his skill and his good theme

45:2-9 Give a description of the queen
These verses present an ideal picture of who the king is.
The king is the best looking of men- vs. 2;  I Sam. 9:2; 16:12; II Sam. 14:25; I Kings 1:6
The king is the mightiest of warriors- vs. 3-5; I Sam. 8:20
The king fights for the most noble of causes- vs. 4
The king loves what is right and hates what is wrong- vs. 6-7

45:10-15 are an address and a description of the king
vs. 10-12 are more of an address to the queen 
vs. 13-15 a description of the queen

45:16-17 A Statement about their offspring, the descendents

Is the king called God? Exodus 21:6; 22:8-9 refers to judges who base their decisions on God's as Elohim. Could a king who bases his authority on God's gift be spoken of in the same way?

These ideals were never fulfilled in Israel's kings and led to an expectation of One who would fulfill these in a way beyond all comprehension.

Hebrews 1:8-9 quotes Psalm 45:6-7 and applies the words to Jesus.
Other things apply to Jesus as well.