You're all set then! However, computer monitors are progressive scan rather than interlaced, so you will actually be playing 1080p, which is a better deal anyway since you'll get double the framerate. The "i" and the "p" are indicators of how the monitor scans the picture, but the resolution is the same.
A standard interlaced scan at 60Hz shows 30 frames of horizontal and 30 frames of vertical every second, so if your game goes over 30FPS, you can't see it anyway. Interlaced, like a computer monitor, scans the full 60 frames per second, meaning that you will be able to see a framerate that high.
The best example I could give is playing Grand Theft Auto 3 on your PC. It has a "frame limiter" feature that cuts the game down to 30FPS (or 25, I can't remember) which was originally intended for televisions. When you turn it off, it makes everything run at a full framerate, which actually looks pretty weird when you're used to the frame-limited game. But it's better!