Well, it's not a hardware problem unless something is broken. You're well over the recommended specs for the game.
I'm guessing there's no error message involved.
If you don't have Steam installed, you need it to run Hitman. Make sure the Steam client is on and then try running the game if you haven't already. Steam often starts automatically when you click on a game exe that uses it, but some games require you to open the Steam client first or they won't even load. It's a bit weird that way.
If that doesn't work try going into properties by right clicking on the games exe file and go to the compatibility tab and check 'run as administrator'. I've heard that Hitman Absolution likes to have administrator privileges and sometimes won't run without them.
It could be something as simple as running the game in windowed mode. Sometimes I've got to start a game in windowed mode the first time I boot it up because the default is a resolution my screen/GPU doesn't like. Once I'm in the windowed mode I can adjust the graphics settings to a more friendly resolution and return to fullscreen. Usually I get a black screen when this happens, but there have been one or two games that behaved like you're describing because of it.
