I am not familiar with photography and have a question regarding the most effective way to get a close up photograph of just a dog's eyes. I would like the photo to have great detail, such as being able to see separate hairs around the eyes as well as detail in the eyes themselves. I am not sure if I'm being too general. I guess I'm looking for the best settings to use, lighting, etc. thank for any assistance you can provide. I have a Nikon D5100. I am willing to purchase a macro lens if suggested.
If you can find a way to make your composition work with limited DOF, then that's great, and this is often the key to good macro photography, but if what you have in mind really needs more depth, read on.
If you can get enough depth of field with a small aperture, then that's the way to go. If you are buying a macro lens, pay attention to how far you can close the aperture down.
If you don't quite have enough DOF, try backing off and using a smaller part of your picture area. You have enough megapixels to sacrifice some.
Another way to do it would be to use a camera with a smaller sensor and decent close focus. Given good enough lighting to use a low ISO, you'll likely get a better result out of a 5 year old canon Ixus for 50 bucks on ebay than you can get out of any macro lens on any SLR, because thanks to the tiny sensor, the DOF is outstanding.
There's some fancy tricks you can do by blending multiple images with different focal distances. These are mostly useful for stationary subjects and a tripod though. Not so useful for a live eye.
I am not a specialist in macro photography, but...
I would recommend a macro lens. Try and do it somewhere with bright natural sunlight (outdoors, or by a door or window). Have a low ISO to keep noise to a minimum, the lower the better.
If it's sunny, don't do it in harsh sunlight - this will give harsh shadows. I'd recommend late afternoon. If it's cloudy or overcast, this will give a much softer diffused light - nice for portraits.
If you're not comfortable shooting in manual, try photographing in Aperture Priority mode and see what shutter speed this gives you when the aperture is at, say, F4. Then switch it over into Manual and use trial and error around the settings until you get the effect you're after. Ideally you won't want the DOF too shallow as you will want to capture all the detail front to back through the frame.
Hope this helps as a starting point :)