The answers here give you reference to the first draft genome. Take a look at these papers for high-quality sequences: Prufer et al., 2014 for Altai Neandertal, Meyer et al., 2012 for the Denisova individual. A third one, the Vindija Neandertal has been sequenced and the sequence is already available (though not published, yet) here. However, one thing has not been mentioned. The presence of ancient DNA can be confirmed by identifying specific damage patterns in the DNA molecules, i.e. mainly deamination. That is used to ensure that the DNA is old and does not result from contamination of present-day humans. Additionally, modern human contamination levels are estimated by two or three independent methods in the papers given above. These methods exploit the uniqueness of Neandertal mitochondria and Y-chromosomes, i.e. the fact that these variants have never been seen in present-day humans.
This answers the sequence part, but my answer focuses on the second part of your question and the issue your raise in the comment sections:
If you say "From their bones/skull" but how do you know they are actually Neanderthals and not Humans with some rare gene mutation or something else?
This can mainly be answered on two different levels. The first level is a parsimony argument: there are a lot of those skeletons (far more than the ones that have been sequenced) and it would be too unlikely they all suffered from the same disease. Additionally, skeletons of present-day humans with the diseases you mention do not at all look like Neandertal skeletons.
The second level is a bit more scientific: bones from the spanish cave Sima de los Huesos have been sequenced as well (Meyer et al. (2014) for a mitochondrial genome and Meyer et al. (2016) for nuclear sequences - not full genome!). Bones from this cave has been dated to approx 430,000 years ago which means that those bones may not even be Neandertal bones but rather Homo heidelbergensis. This makes a strong point against your suspicion as at that times modern humans did not even exist anywhere on the planet and it took another about 200,000 years until the first anatomically modern humans evolved in Africa (and then another 150,000 years until they set foot to Europe).