Continuity mistake: After the scene showing the First Order ships appearing behind the Resistance ships, when Poe Dameron says "It's Snoke's ship. You gotta be kidding me!", the immediate next shot shows him further away and facing a total different direction from Admiral Ackbar. (00:26:30)
Continuity mistake: When the light saber breaks in half, Ben is thrown to the left of the throne while Rey is thrown to the right. Once Ben wakes up, he is on the right of the throne, exactly where Rey was when she refused him. (01:51:50 - 01:55:10)
Continuity mistake: When Rey meets Luke for the first time, the sun changes between shots.
Continuity mistake: While Rey is practising with the pole and the lightsaber, the sun changes between shots.
Answer: As he says, "let the past die. Kill it, if you have to." Han was his past - he didn't hate his father, but his existence was holding Kylo Ren back from reaching his full potential, or so he believes. The principle is demonstrated earlier in the movie when he can't bring himself to kill Leia, but has no issue with the other TIE pilots blasting the bridge and (he thinks) killing her. He wants to free himself from the shackles of his parents, who cause him such internal conflict. Remove the source of the conflict and he believes he can move on to greater things. Of course, arguably his position is a bit naive, and his actions will actually cause him greater problems down the line.