In interviews about the character, people involved with the show describe the character as bisexual.

In our time like the 20th or 21st century, when being proud and non-heterosexual is enough to rattle people's sensibilities and make them question their assumptions, Jack would accept that label.

But in actuality, he's omnisexual: being from the 51st century, he is down with any gender of many humanoid species. The Doctor mentions in the episode The Doctor Dances that this is accepted human sexuality in the 51st century.

Captain Jack: "I'll disctract [the male guard]. Don't wait up."

[Jack runs off to seduce the guard leaving a suprirsed Rose.]

The Doctor: "Relax. He's a 51st century guy. He's just a bit more flexible when it comes to dancing."

Rose: "How flexible?"

The Doctor: "By his time you lot have spread out across half the galaxy."

Rose: Meaning?

The Doctor: "So many species, so little time."

Rose: "What? That's what we do when we get out there? That's our mission? We seek new life and, and ... ?"

The Doctor: "Dance."