Is this character really a bad guy?

While he may not be the main antagonist in the scheme of the film. That role belongs to the Admiral for all his misdeeds during the flow of events.

Khan is still an antagonist no matter how you look at it.

Back in the 1990's, the Eugenics War occurred where it was a battle for superiority of normal humans vs these seemingly indestructible genetically modified super-humans. In the end, the super-humans lost and because of their rebellion, were sentence to float about space for the rest of their time in a cryogenic state, including their leader of the rebellion, Khan.

When the Admiral released Khan, his intention was for Khan to help ignite the Klingon war using his designs for warships like the Vengence. Meanwhile, Khan only agreed to the Admiral's demands because he knew the Admiral would kill the remaining super-humans if he didn't comply. All the meanwhile, Khan is putting his plan into place to get his comrades free and re-ignite the Eugenics Wars using this new technology of the era.

The only reason Khan complies with Kirk's demands is because he learns that his comrades will be destroyed when these missiles detonate upon impact. Which is also the only reason he saves Kirk as the warning was sent out to him long before the land party departed, alerting Khan to the fact that his old friends might be in danger.

From then on it becomes Khan using the crew of the Enterprise and make them sympathetic and curious as to the situation at hand, which is why the Admiral wanted Kirk to just fire the torpedoes and be done with it.

Khan then uses Kirk to get aboard the powerful war vessel that he helped the Federation design and once he is able to subdue everyone on board, unleashes literal hell up on the crew of the Enterprise with his intended target Earth, after getting his comrades.