They do not sound "very weird." They're perfectly correct (except for one small mistake). In real life, in spoken conversation they would sound fine to a native speaker.
You should say "I won the championship." A "champion" is a person (or team). The "championship" is the thing that the champion wins. After winning, a "national champion" is officially regarded as the best in the national, the "league champion" as best in the league, and so forth. "Championship" has two related meanings, one very abstract. It can mean a game or (usually) series of games, to find out, officially, who's the best of all. The abstract meaning is "the state, position, condition, or title of being champion."
In the United States, "do some sports" is natural, colloquial, spoken English and does not sound weird--but "play some sports" would be better.
Since you are a woman, according to your profile, I agree with Lokesh that "buddies" doesn't sound quite right. That's because I associate it with male bonding, and very close male friendship. I'm old and probably out of date on that, though! Neither ahdictionary.com nor oxforddictionaries.com says anything about gender. "Buddy" is actually a variation of "brother," probably based on one way little kids often mispronounce "brother" when learning to speak. I would play it safe and say "hang out with my friends."
"Allowed me to gain a great sense of achievement" sounds tentative. Literally, it means there was something stopping you from gaining that sense, and winning the championship removed that barrier. I would just say "winning the championship gave me a great sense of achievement." There's a slight difference in meaning between "allowed me to gain" and "gave me." "Allowed me to gain" sounds as if you're choosing words very carefully, like an academic, and thus sounds formal.