i used to have a nikon coolpix, just recently got a canon but i think these tips would help with any digital camera.
- use the macro mode (usually represented with a little flower button)
- sometimes, flash washes out the colour so try to take your picture near a natural light source (window, outside, etc)
- otherwise, if flash is really needed: stretch your arm out as far as you can (or mount the camera somewhere) and zoom in. that way, the flash isn't as close to your skin and won't wash out the colour
- if you have shaky hands, set the timer to two seconds. in case you press the shutter too hard or move the camera while you're pressing the shutter button, you have two seconds to try to steady it
- if you're taking pictures indoors, try setting your white balance to tungsten or fluorescent to get the colour right
- take your closeups with high res. that way you can crop the picture in an image editing software to just focus on your eyes if you wanted to, but still have a pretty big image
i hope those tips help! those are all i can remember for now.