When you do something even though you really don't want to, you do it begrudgingly. If you hate to admit that your brother's cookies came out better than yours, you'll only do it begrudgingly.
To act begrudgingly is to begrudge, a word that's rooted in the Middle English grucchen, "to find fault with or be angry." So when you find yourself grumbling about something (but doing it anyway), you're doing it begrudgingly. If you dislike asparagus, you're only going to eat it begrudgingly, and if you're bitterly jealous of your friend's award, you'll applaud her win begrudgingly.