The first thing you have to do is to sequence the Neanderthal genome, and that has actually been done. The next step would be to chop this genome up into, say, 10,000 chunks and then... assemble all the chunks in a human stem cell, which would enable you to finally create a Neanderthal clone.

If society becomes comfortable with cloning and sees value in true human diversity, then the whole Neanderthal creature itself could be cloned by a surrogate mother chimp - or by an extremely adventurous female human.

The one thing that is bad for society is low diversity. If you become a monoculture, you are at great risk of perishing. Therefore, the recreation of Neanderthals would be mainly a question of societal risk avoidance.

George M. Church Quotes