One way to get around the seepage problem: presumably, your monitor looks about like this:

 ______________________
|                      |
|   _________________  |
|  |****************|  |
|  |****************|  |
|  |****************|  |
|  |****************|  |
|  |****************|  |
|  |****************|  |
|  |________________|  |
|      . . . . . .     |
|______________________|
      /         \
     /___________\
(the asterisks being the screen, the periods being the buttons (usually along the bottom).

So, the parts around the screen are what you want to paint. Do the bottom first, with a strip or two of masking tape along the bottom of the visible portion of the screen. Make sure you have newspaper under your monitor, so you don't drip on the desk! Wait for it to dry, then rotate your monitor 90 degrees along its Z-axis as it sits on your desk. Then, paint the right side of the monitor the same way (with masking tape also). Repeat for the top, and left side, rotating the monitor 90 degrees each time. Being rectangular, it should sit basically level each way. After the last side has dried, rotate it back rightside up, and repeat the process for a second coat. Since gravity should draw the paint downwards, towards the centre of the earth, it will be much less likely to seep under the screen.