Sure you can!
People do it all the time. To use your own example of Abraham Lincoln, there was a film from a few years ago (adapted from some other medium) depicting him as a part-time vampire hunter. It was imaginatively titled Abraham Lincoln: Vampire Hunter, wove fictional events together with real events, and to my knowledge, nobody from Lincoln's estate had a problem with it.
Then there's Futurama, in which Richard Nixon's head is the President of Earth; Doctor Who, which has had Churchill recruiting Daleks and Agatha Christie fighting evil space bees; the obscure but very good anime Nobunagun, in which
Jack the Ripper was actually Florence Nightingale and
his her victims had been infected with alien parasites
...you get the picture.
You've said you intend to maintain the "reputation and natures" of the people involved, which is probably the most important part. As I commented on Surtsey's answer, you'll want to be careful of libel. The further back in history you go, the more leeway you have, as people lose connection to their ancestors over time. When Shakespeare wrote his history plays, he had to be careful to portray the then-ruling Tudor dynasty in a positive light, or he would have been locked up. When he wrote Julius Caesar, however, he could do whatever the heck he wanted; Caesar had no direct descendants to rock up and say "No, that's not what happened".
In short: as long as you're not actively demonizing anyone who doesn't deserve it, go crazy! Have fun! Make Abe Lincoln fight alien space bees if you want! (Seriously, I'd buy that book.)