Did the average imperials know that they were the "bad guys"?

Generally no. They probably don't even think that they're the "bad guys". In fact, most of them probably don't think about the matter much at all and are probably more interested in just getting on with their lives regardless of who's technically in charge.

There's also evidence (prevalance of smugglers, Hutts who maintain privately-operated fortresses where they play by their own rules, bounty-hunters who the Empire pays rather than commands, well-armed and generally lawless populations that tend to significantly outnumber Imperial forces) that many people living within the Empire, and particularly on worlds along its fringes, see it as more of a nuisance to be avoided than something that they actively support, participate in, or oppose.

Outside of its struggle against the Rebels, there's fairly little evidence to suggest that Imperial rule was overtly oppressive to anyone who didn't actively fight back (you can have your blaster, get drunk in the local bar, hassle the newcomers for no good reason, and get your arm sliced off and no one will bat an eye, let alone bring Imperial forces in to help or hinder your activities). And not much stated about how much news about the Empire's military atrocities actually filtered back to the general population. Also bear in mind that a good proportion of the Imperials who are actually participating on the front lines are clones conditioned for obedience.

So the average sentient being living on an Imperial world and not participating in its military would probably not think they're on the "bad" team. They'd probably buy Palpatine's narrative about an ongoing war that started out with Trade Federation separatists, progressed to a near-successful assassination attempt by members of the Jedi Council (who most average beings probably cannot tell apart from the Sith that he's had publicly fronting the separatist army; given that both are deadly warriors with mystic powers and lightsabers) that left him horribly disfigured, and is simply still ongoing.

I imagine his propaganda would play well with anyone who doesn't have detailed insider knowledge about what was actually going on. Doubly so since for all their wisdom it doesn't seem like the Jedi ever mounted anything resembling a coordinated counter-information campaign of their own. Triply so since most would probably not really care that much as long as the conflict is not affecting them personally.

If they don't consider the Empire to be evil, then how can they justify the Empire's action to themselves?

How much do they know about the Empire's actions? You never see any journalists embedded with storm-trooper deployments.

But in any case, I imagine they'd justify it in roughly the same way we Westerners justify the number of innocent men, women, and children that our activities in the Middle East have killed. Collectively, as "collateral damage" in a time of war, and as something that most people decry, accept, and then forget about almost simultaneously.

Of course, you'd think the destruction of an entire planet is something that people will find out about, and a large enough atrocity that it can't simply be dismissed as "people die in war". Which does seem to be the case:

Once word of the world's fate got out, thousands of Alderanians who had been off-planet joined the Rebel Alliance. Alderaan's survivors became among the most dedicated soldiers of the Alliance, vowing never again to let the Empire destroy another world. Most Imperial military members from Alderaan immediately defected to the Rebellion.

...but also:

A few of the Alderanian survivors became supporters of the Empire, blaming the inept and traitorous first chairman Prince Bail Organa and the rebellion for opposing the Emperor and thereby destroying everything they had, even their loved ones.

So it seems the threshold for justification is when the Empire's evil becomes personally relevant. Which, from the standpoint of human history, is unfortunately accurate. Most people have shown themselves willing to shrug off any perceived or real evils that their nominal nation-state might inflict, so long as those evils are being inflicted upon someone else.

It's only the exemplars and the people with insider knowledge (like Padme, Bail Organa, and Mon Mothma) who will both recognize and take a stand against evil even when they don't personally have any skin in the game yet.

I think this is a case of art imitating life.