他人任せ ('leaving it to others') carries a negative, shirking nuance. The clipped ~ないと is a colloquial abbreviation of ~ないといけない/~ないとダメ ('have to / must') — a very common spoken contraction. ~てあげる adds the benefactive direction: the speaker's action is done for someone else's benefit. 俺自身 ('I myself') doubles up on the first-person pronoun for emphasis, driving home that no one else is going to do it.