For example, "How much" or "What" is your chess rating? Which one is correct? Also, can I say "You may improve your chess rating by winning games." or "You may lower your chess rating by losing games."?
Let's first address your last two examples:
You may improve (increase/raise) your chess rating by winning games.
You may (decrease/lower) your chess rating by losing games.
These are both grammatically correct. Some of us might prefer "could" or "can" rather than may", but others of us might not, lest we imply that the hearer is capable of winning more games.
As for "how much " vs. "what", that depends on whether the rating is on a point system. If it is numeric, you can ask "what" OR "how much", but if it is non-numeric, you can only ask "what", not "how much".
However, in chess there are (numeric) ratings, (ordinal) rankings, and (named) titles. For example, one player might have a rating of 2400, have a title of Grandmaster, and be ranked 4th among females under 21.
So you could ask "How much is her rating? What is her title? What is her ranking (or how does she rank)?"
So, speaking of "increasing": I would say increase your rating, improve your ranking, or attain a higher title by winning more games.
'Rating' in this context is a measure of how good you are e.g. "Messi is often rated as the best football player in the world.". It may have an actual, measurable figure (in golf, rating is expressed as 'handicap'; Tennis uses 'ranking').
It can be used as a noun or a verb e.g.
"How do you rate England's chances in the World Cup?"
"What's Ronnie O' Sullivan's snooker rating?"
"I see he's moved up the chess ratings with that victory."
In the first example, 'rate' is used as a verb, meaning "How well do you think they will do?". In the second, 'rating' refers to his individual rating or measure of his skill and in the third, 'ratings' refers to a table of all the players ratings and the fact the he is higher in the table now, as his individual rating has increased.