The real key is to find a way to commit your timeline-corrective murders while you yourself are still very young in the second run through. That way it's all juvenile crimes on your record!

Alternatively, you could probably change the timeline without resorting to homicide. For example, let's say you don't want someone to become president. Send them a hundred copies of a letter. On that letter, state that you are a psychic, and that you are writing them to give them a warning. You tell them the future can still be changed, but on their current path, they are headed for a cruel end. You then list a bunch of future events, including some from their life. And then you include one final one at the end.

For example, let's say you wanted to prevent Trump from getting elected president. Send him a letter in 1996 telling him the future. Describe all sorts of future events. Maybe even throw in a few he can make a little bit of money off of. Hell, put in some stock tips. You want him to think you really have his best interest at heart. But then, for the final prediction, state that on the current timeline, he will run for president in 2016 and be assassinated on the campaign trail. Then provide a list of future calamities this apparently caused. Tell him that if he declines to run in 2016, then all sorts of good things will happen.

Hitler killed himself. Go back and kill Churchill.

The problem is, most of the likely targets are front-men puppets, and the puppet-masters hide in the shadows. Finding who to optimally eliminate (or otherwise redirect their life, without murder), to mend the world, is no easy task. It's murky at the "top".