Both work depending on the context. Normally, you'd say "I'm Anderson, just a guy who's trying to improve /my/ English." What you really mean is "I'm Anderson; [I'm] just a guy who's trying to improve my English. However, you can talk about yourself in the third person. Not that many people do this, but some people do; it's like they're describing their own story. When you say "I'm Anderson, just a guy who's trying to improve /his/ English," that's the vibe you give off. (At least in Los Angeles) For example, you might hear someone say, "Yeah, I'd like to think I'm a pretty cool guy who minds his own business and keeps his nose out of trouble." In this case, you're describing the "pretty cool guy," which is why you speak in third person, but in the beginning of the sentence you lump yourself into that category. I don't know how much they do this in the US, but in Los Angeles it's somewhat common. Hope this helps!

MY is the correct answer. You introduce yourself in FIRST PERSON - I am Anderson. So you need to stick to first person throughout. Alternatively use 3rd person, Anderson is just a regular guy trying to improve his English. Alternatively try 4th person, and talk to the audience, like Deadpool does.

Guess you should say "his" because you use "... who IS trying to ..."