5 and a half hours in I got this from Amanda Holliday... oh lawdy.
EDIT: So to test out this bounty, just got 19 kills with a 2.71 KD with TLW (plus a legendary chest engram!), which gave me 19% completion. So seemingly, 100 kills with TLW equipped is for Part 1.
EDIT 2: Seems like at this point most people who finished the main story quest got this bounty. At the time I made this thread pretty much nobody had it, so please stop replying saying "Its not rare" or "Everyone has it", we know.
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1 point for normal kills 2 point for precision kills -1 point for death Some people have reported that they occasionally lose 2 points on death though.