That story line is centered around some the characters in the King of Fighters video game series. Kula (the girl) and K'(the guy) were both created by a organization called NEST in order to kill Kyo Kusanagi. Ironically, both of them actually gained their powers through Kyo, as they are nothing more than genetically engineered clones of him. K' refused to continue his plan, so Kula was sent to eliminate him.
So thats why Kula is trying to kill K'. As for why she ran away; she also doesn't like working for NEST. The only person that she actually tends to trust is the character Foxy(I think she was mentioned...)
As a side note, the weird guy in the beginning of the story was K999 (i think), who is also a clone of Kyo but has some weird powers. He, along with his partner Angel, is devoted to the NEST organization and ends up killing Foxy, leaving Kula is a difficult situation.
Did that help?
Asobi Wa Owari Da! Nake, Sakebe, Soshite Shine!!