Warning! Spoilers from Star Wars VII: The Force Awakens... I shouldn't have to warn you about spoilers, because you should've seen it by now, but... Ok, this is my short theory on Kylo Ren.
What DID Darth Vader start?
Before he died, he says to Luke, "You were right. You were right about me. Tell your sister... you were right..." Meaning that there still was good left in him, and that he realized he was good after all, yet he chose the wrong side.
My question is... Will Kylo Ren actually finish what he started?
If Darth Vader was good after all, will Kylo try to be an undercover spy-like role in the Star Wars movies? Will he go behind Snoke's back and act as if he is on the dark side, and then turn on the dark side, and help the rebels and what is left of the Jedi (Luke and Rey, and possibly Finn), destroy the First Order/dark side once and for all in the next couple of movies??
Then there are the questions like, Why'd he kill Han and What did Kylo Ren mean by 'will you help me' when he killed Han and did Han know what Kylo meant by "will you help me'?
Here's what I think...
When Han Solo encounters Kylo Ren in Star Wars VII,
Han Solo: Ben!
Kylo Ren: Han Solo. I've been waiting for this day for a long time.
Han Solo: Take off that mask. You don't need it.
Kylo Ren: What do you think you'll see if I do?
Han Solo: The face of my son.
Kylo Ren: Your son is gone. He was weak and foolish like his father, so I destroyed him.
Han Solo: That's what Snoke wants you to believe, but it's not true. My son is alive.
Kylo Ren: No. The Supreme Leader is wise.
Han Solo: Snoke is using you for your power. When he gets what he wants, he'll crush you. You know it's true.
Kylo Ren: It's too late.
Han Solo: No, it's not. Leave here with me. Come home. We miss you.
Kylo Ren: I'm being torn apart. I want to be free of this pain. I know what I have to do but I don't know if I have the strength to do it. Will you help me?
Han Solo: Yes. Anything.
Kylo Ren ignites his cross-guard saber into Han Solo, impaling him.
So, Kylo's being torn apart with the decision to kill his father to help Snoke believe that he's really on the dark side, so that he can sneakily destroy the dark side/First Order from the inside out. That's why he asks his father to let him kill him... in a weird, incomprehensible way... But Han may not have deciphered all of this to know what he meant, although he does touch his cheek in a longing, fatherly manor, he may or may not know why Kylo kills him, just as we have no idea, leaving us to form these crazy concepts and theories!