The most humane and viable manner in which you might do this is simply start a religion in a Bene Gesserit fashion where the prophesied Kwisatz Haderach is a Neandertal. Over the generations by selecting individuals within the population (which could draw in converts) enriched for Neandertal ancestry to mate assortatively one could slowly increase the proportion of that ancestral component.The idea is simple: if you mate two persons with x% and y% Neandertal ancestry, you will get (in the expected sense) a baby with 0.5(x+y)% Neandertal ancestry. In actuality the baby's Neandertal ancestry will be a random variable z with mean 0.5(x+y) and some variation around that mean.
Now, if x and y are very different from each other (e.g., at the limit you had crossed a full Neandertal with x=100% and a full sapiens with y=0%), then z will always be in-between x and y.
But, if x and y are fairly close to each other, it's possible that thanks to the wonders of recombination it will happen that z will be greater than both x and y, because the offspring might actually happen -by chance- to inherit many Neandertal alleles from both mom and dad.
Project the above procedure over the millennia, and it's easy to see how one might incrementally increase the proportion of Neandertal ancestry in the gene pool to an arbitrary degree: individuals might start with, say, 2% Neandertal ancestry, and given enough crossings per generation (so that enough individuals with >2% Neandertal ancestry will be produced), and enough generations, the number will inch along, and converge to 100%.
But, actually, reaching a 100% Neandertal might be impossible. To see why, consider that while many Neandertal alleles introgressed into the modern human population a few tens of thousands of years ago, surviving down to the present, some of them did not introgress and some that did were culled away by either drift or natural selection.
Moreover, we might not be able to utilize all extant Neandertal alleles because we can mostly identify Neandertal-ness with respect to ancient DNA samples.
In other words, the modern human gene pool has a lot of Neandertal alleles floating around, but some of these don't look like Neandertal alleles (because we didn't find them in any ancient specimen yet), while others are simply long gone.
Such a breeding program would therefore not produce a Neandertal in the end, for the same reason that you cannot produce a grey hue by mixing different shades of red and blue: you also need some yellow. The modern human gene pool may have lots of Neandertal alleles but it may not have enough.
In any case, it would still be possible to create a, say, 50% Neandertal, or whatever the proportion of Neandertal alleles still in existence is. But, will such an individual be a Neandertal in the physical sense?
Which were the alleles that conferred Neandertal-ness to their bearers? We don't know, but this property may very well have involved both (i) combinations of alleles that exist in modern humans, and, (ii) alleles we no longer possess.
As an analogy, imagine that in the future, polar bears are extinct, but some of their DNA introgressed into other bear populations. Future zoologists might breed specimens that have a lot of polar bear alleles, but will lack the alleles conferring white coat color. Such a hypothetical breed of bear might possess a high degree of genome-wide similarity to actual polar bears, but it won't be the same where it counts.
Thus, it may very well be possible to create individuals from extinct human populations (be they archaic hominins such as Neandertals, or even admixed modern humans such as Tainos), but one might not get a 100% replica of such populations by breeding alone. Even if one was able to reconstruct a great part of such creatures' genome, the most important portions might still be missing.