Let's take this in steps - as should be with the issue at hand.
Should he be extradited and tried?
That's TWO questions :). First is the issue of extradition. In this case the answer would be "yes" IMHO, but not for the reasons he is currently extradited for. I vehemently disagree with the notion that someone should be extradited without a review if the alleged evil deeds are a crime in the country of residence and if the evidence so far stacks up to make sure it was actually him who did the hacking. Unfortunately, this is what one T Blair agreed to when he was busy pretending to be George W Bush's poodle.
What Tony Blair agreed to is simply wrong because it allows another nation to impose its laws on the UK, impairing UK sovereignty. If that wasn't a problem, Brexit would not have happened either - you can't have it both ways so this T Blair agreement need to be annulled - but that's a separate battle.
In other words, I agree with the extradition as his alleged crimes happen to be to be criminally actionable in the UK as well, NOT just because the US says so.
Is Asperger's a "get out of jail free card"?
No, not at all. Asperger's doesn't impair the ability to tell right from wrong. It's a convenient excuse because it's a less understood condition buried in myth and, frankly, FAR too often abused as excuse. People with Aspergers do not commit crimes by default, thank you.
Did he or did he not break into the computer systems?
THAT would have been one to be worked out by UK courts prior to extradition instead of bending over by default to the US legal system and shipping someone out. If you want to talk Human Rights, well, there's your work - he is entitled to decent due process, not to a media circus driven kangaroo court with a number of new players soon to be elected by someone with orange skin, weird hair and an at best tenuous relation with reality.
Does it matter that it's US systems? What if it were another country who's computers he broke (allegedly, of course) into?
It SHOULD not matter, but it does. As it's the US, it's "no questions asked" and the chap gets crated and shipped as soon as everyone has made their money through the various plays in UK court, whereas the whole process would be a bit more thorough if it came from anywhere else, sans Tony Blair negotiated bypass of UK law.
I believe anyone who breaks into a computer system should be tried in the country where the computer(s) were/are located.
Yes, but it's difficult. Try extraditing anyone from Russia, for instance. You'll be thanked for given them something to laugh about, but it won't happen. Furthermore, there's no reciprocity either: try extraditing an American who has broken into UK systems. That, IMHO, is one of the key arguments why the US ought to be told to f*ck off in the first place until full process has taken place - symmetry and balance matters.
But... I'm curious how others feel as I have no real answers on this
It's not quite as black and white as the press would like to make it appear (well, it wouldn't fit in a 200 word article then, of course), there are quite a few variables in this which is why I tend to take the time to unravel it all before forming an opinion. I know that my absence of snap outrage isn't fashionable right now but I prefer to form my own opinion, on account of having a reasonably well functioning mind (post coffee).
In my opinion, if you can't handle doing the time, don't do the crime or man up and face the consequences. The way the UK handles extraditions to the US is simply not right and it ought to be changed, but I think that in the circumstances here the outcome would have been identical, even with the T Blair agreement annulled as it should.
Sorry, long post, but I have tried to untangle the various elements involved - exactly because there are plenty who seek to do the opposite.