You can connect an iPad to a monitor, either using AirPlay (as suggested above) or a Lightning to _______ (VGA, DVI, HDMI) adapter (most of which come in two flavors - just the port or the port + lightning so you can charge while projecting).
The hiccup is that you still need to use touch on the iPad (there is no mouse support) to navigate and click (a Bluetooth keyboard can still be used).
These solutions are meant more for projecting for an audience and not so much as using as a desktop PC.
However, you could have a Mac (or PC) connected to the display and use the desktop computer to remotely control the iPad - in which case you can totally use the mouse and keyboard. You would be okay with touch points that require a click, but anything requiring gestures are harder to pull off with a mouse.
In short, it's not really possible to use the iPad as you suggest at this time.