It really depends on the ressources I would be allowed to take with me. And of course on the realism of the assumption. Travelling back to the roman empire would probably wipe out 80% of humanity, considering the amouont of modern germs present humans carry around with them - and it also could mean a speedy death at the hand of some forgotten plague.
If I were allowed to bring some equipment with me, I'd bring a solar-powered battery charger, a digital camera, a laptop and some aspirin, someone with knowledge of arameic and greek and latin, a lot of gold coins and some guns, and would probably attempt to copy every scroll of the library of Alexandria I could lay my hands on. Securing the complete works of Aristotle would be enough, but salvaging a couple of thousands of scrolls, or ven more, from this lost treasure of human wisdom would not only make me rich, it would emply thousands of translators, historians, and philosophers for decades. It would change and refine and alter our view of the past, o philosophy, and solve a lot of historical puzzles.
Probably the burning of the library in 44BC was such a botched attempt, hm? :D
The second intervention I would attempt is not to "Kill Hitler" but comes close: Travel back to 1914 and create enough havoc that the Germans win the Battle of the Marne. the result?
- No trench Warfare on the western Front, millions of people survive
- No long fight on the Eastern Front, either, therefore:
- No communist uprising (Lenin stays in Swizerland), sparing 60 million lives and bringing Russia into modern Europe
- Germany creates and leads an european "Zollverein" (customs union), which today is called EU
- France and Britain loose their world power status, sparing them a couple of ugly colonial wars
- Hitler stays a failed Austrian painter
- The rise of the US to world power gets slowed somewhat (cause Europe doesnt have to borrow itself senseless)
Of course, all speculative, everything can still turn out terribly. ut it would sure be fun to try :D
Or I would just go back a couple of years and kick myself in the butt repeatedly fo breaking up with my ex.