Thank you, and yes, he does remember his ancient life--he's still the same man--trying to fit in, and faced with all the problems of our modern world.
Well, then I would have a little problem believing in this "science" in your fiction. It is rather basic scientific knowledge that DNA can't store living memories, it is our brain domain. As much as I like science fiction I would have a hard time getting over this premise.
My Alexander is a little more tame than that, because today's world wouldn't tolerate his aggression, but over the course of the three sequels I've written, he does rise to prominence again.
You might have a psychological problem here. If he remembers all the past, his past behaviour, actions, and also aggression, likings, love, hate, excitement,bravery, domination, killings etc, then how come he can be tamed today? Having a copy of real Alexander with same character traits and with all his memories you wouldn't be able to make him a different man. If you did he would have been very unhappy and depressed living as an ordinary man, just reminiscing how bravely he sent thousand men to death on battlefield and gang raped whole village of women, and not only, if the whole legend of him is right. Also possibly, growing in today's tamed, peaceful and feminine world, he would feel extremely guilty of his actions of the past life, maybe getting suicidal often. Heck, he might have been prosecuted for genocides, murders, rapes, arson of whole cities, slavery, cruelty to animals, the list goes on.
There is also another unforeseen consequences of withholding memories of past life. Knowing that this is possible, scientists would clone other figures from past to gain knowledge of past events, rewriting the world histories and deromanticizing our past and legends.
What if someone brought Genghis Khan to life? Would he love to fight Alexander to finely figure out who was the best warrior in the world? Maybe it would culminate in Armageddon?
It's a bit of slippery slope, or Pandora box you opened.
I really would change the memory in DNA in science fiction to fantasy realm, and lets say, that the Spirit of Alexander came back to his cloned body, thinking correctly that this is the same Alexander (genetically speaking) and brought memories of past life back with it.
Sorry for "critiquing" to much, but I was in the mood.
All the best.