Darren Criss is making a (TV) murderer.

After years of being known as preppy singer Blaine Anderson on Glee, the actor takes a dark turn in American Crime Story: The Assassination of Gianni Versace as Andrew Cunanan — the serial killer known for shooting the Italian fashion designer on the steps of his Miami mansion and murdering four other men in 1997.

“I had a friend tell me when I got the part, ‘You’re playing the gay boogeyman,’” Criss, 31, tells PEOPLE in this week’s issue, on newsstands Friday. “I was like, ‘Excuse me?’ He was like, ‘When he was on the run, we would all spook each other [by saying] Andrew Cunanan is going to come get you.’ The things that are said about him in the show aren’t crazy.”

When it came to playing Cunanan, Criss wanted to make sure he was portraying all of the complex aspects of his character, which included everything from killing in cold blood to singing Laura Branigan’s “Gloria” at the top of his lungs in his car.

“Human beings are so complex,” he says. “We are capable of so many different emotions and the reasons behind those emotions. I’m not asking people to empathize or pardon anything that Andrew has done, but I do like people unconsciously figuring out how much they can relate to this person whether how little or how much.”

He adds: “It is my job to humanize him, but the hope is that we’re not glamorizing anything.”

The series, which is based on journalist Maureen Orth’s book Vulgar Favors, starts with Cunanan’s murder of Versace and moves backward through his killing spree. While viewers have now seen how all of Cunanan’s murders went down, Criss promises that scenes of his younger years are soon to come.

“That was some of the most fun stuff for me,” he says. “With someone like Andrew, I don’t think he had homicidal tendencies as a teenager. He was a lovable, fun, smart, gifted kid and it is confusing and heartbreaking and mortifying for people that knew him at that age to think he’d be capable of something like this later.”

Before filming, Criss says he knew just as much as “most people tuning in” about Cunanan — basically only that Versace was shot by a young half-Filipino man like himself.

“I met quite a few people who didn’t even know Gianni Versace was murdered,” Criss says. “What I’m getting to realize is that, for the most part, people didn’t know a whole lot about Andrew.”

Since taking on the role of Cunanan, Criss says people started to come up to him to share their stories about him.

“When he was alive, he literally was everywhere in the sense that he knew people, people knew him and he made himself the life of the party,” Criss says. “Even after he gained this degree of infamy that augmented his persona, people would have stories about him or think they saw him.”

Though the show has been met with its fair share of controversy — the Versace family slammed it as a “work of fiction” in January — Criss says he can understand why.

“I don’t blame anybody for having any reaction to this,” he says. “I mean, that’s their family member on TV. It’s completely understandable. You just hope the work speaks for itself and some good is brought through this.”

For Criss — who announced his engagement to his longtime girlfriend, Mia Swier, on Jan. 19 — the show’s success couldn’t come at a more exciting time.

“Some actors have to wait a lifetime for this kind of stuff,” he says. “This happened exactly when and how I would have liked it to happen in my life.”

He continues: “I just hope I don’t blow it from here!”

Darren Criss on Playing Gianni Versace’s Murderer: ‘It Is My Job to Humanize Him’