How should I set up a camera for monitor my baby?
For a cheap and dirty solution, you can get a light bulb socket that contains a power outlet, e.g., here. If the camera uses a USB for power, you could try this DIY solution.

If you want it neat and tidy, then put in a new power outlet on the wall, either routing the power cable through the wall (requires identifying a suitable place from which to pull the cable: may be relatively easy if you have a basement or attic from which you can access the closet walls above/below), or over the wall in a conduit.