In the film Star Wars: The Force Awakens, Kylo Ren
kills Han Solo, who we found out is his father.
What is the reason for this? Does this act make Kylo Ren more powerful? Would this make him more powerful than Darth Vader?
Kylo Ren has been struggling to maintain his commitment to the Dark Side. When speaking to Vader's mask, he says:
Forgive me. I feel it again. The pull to the light. Supreme Leader senses it. Show me again, the power of the darkness, and I will let nothing stand in our way.
Just before killing
Kylo Ren says:
I know what I have to do, but I don't know if I have the strength to do it. Will you help me?
The obvious implication is that he believes this act will bring him closer to the Dark Side. It destroys a key connection to his former life, and would make it much harder for him to go back even if he wanted to.
Whether this will really make Kylo Ren stronger remains to be seen; the important thing is that in the moment, he believes it to be true.
Additionally (feel free to combine this with @Royal Canadian Bandit's answer, I haven't been around long enough to simply go and edit others' answers, matter of courtesy):
In TFA, at around 50:38
Snoke:The droid we seek is aboard the Millennium Falcon in the hands of your father, Han Solo.
Ren: (after a pregnant pause) He means nothing to me.
Snoke: Even you, master of the Knights of Ren, have never faced such a test.
Ren: By the grace of your training, I will not be seduced.
Snoke: We shall see. We shall see!
Maybe it's just me, but that is foreboding and setting up the tragedy in a well-nigh Greek fashion! Snoke told Kylo that this is a massive test he had to pass. Kylo wanted to be bad so badly, he had to prove it to his boss and himself that he really could be.