I am looking for a word or phrase that means that you have a feeling or need to improve or enhance yourself just for the sake of it.
I think "self-improvement" conveys that pretty well. "Self-enhancement", as well. To express the "need for" element you could add "drive" to the end of either one of those hyphenated alteratives.
E.g., "Joe has an incredible self-improvement drive; he's always challenging himself and doing something new. Unfortunately, he's really pedantic and goes around saying, 'I'm not in competition with anybody but myself.' All the time; it never stops".
And, by the by, "social comparison" is what we say in psychology for process of gauging how well you're doing using others as a measuring stick.
Your question is vague, but the phrase "for my own edification" comes to mind. The gist is, "tell me for no other reason than to allow me, and only me, to improve myself."
If something is for your edification, watch out! It means it's designed to improve you in some way, whether morally, educationally, or spiritually.
Excel, e.g. She likes to excel in everything she does.
Get ahead in the world, e.g. As a young man, So-and-So had a real drive to get ahead in the world. (Note, this is a pretty vague expression, but I think it relates more to business than to academics.)
Do your best, e.g. I'm not overly perfectionistic or anything, but I do like to do my best.