I've been watching a lot of Grover clips lately, and began to see just why he's such an enduring character. He doesn't know how to do anything right, but he's got a lot of common sense.
What he lacks in brain or brawn he makes up for with ambition, honesty, respect, and long, shaggy, open arms. It is easy to think that Grover has a lot of confidence because of all of this, but he has more doubts than a character like, for example, Roosevelt Franklin.
If you want to see Grover in his purest form, look back to the times he tried to help Kermit the Frog with his lectures. Even before he became an established character, you can see what I mean about Grover being very friendly, eager to help out his fellow Muppet friends. When given a task too difficult for him, he begins to realize, and sometimes worry that he doesn't have the proper knowledge to go through with it. That doesn't stop him from trying, for he knows trying is the only way to make progress. He works and slaves for his friends or customers, hoping to eventually achieve his goal, but he only ever gives up if it starts to wear on him too much, either mentally or physically, but usually the latter.
Like most of Frank Oz's characters, you may sometimes see Grover acting a little dramatic or sarcastic, for comedy's sake, but he never means anything ill. Even when Grover gets frustrated with himself or whoever he may be interacting with, he only wants what's best for himself, and for anyone.
If any of you have anything else to add or correct, that would be fine. This is just my interpretation of the character.