To me, it seemed like Quaritch (in the theatrical) strived to kill Jake because Jake was the cause of everything that happened after destruction of the trees of voices.
The way I figure it, Quaritch assumed that even though he had completely failed his original intention to destroy the Tree of Souls, he thought that if he killed Jake, the leadership would be gone from the Na'vi forces, and the RDA would have a much better chance of regrouping at Hell's Gate to attack again (assuming there were still many Samsons unused back at base). In this theory, we could say Quaritch wasn't only thinking about himself, but rather of the RDA's control over Pandora.
...Plus he was friggin' angry that his shuttle was destroyed.