Kinder is used for a simple comparative or description e.g My friend is kinder than me. I wish I were a kinder person. It is possible to say 'more kind', normally to emphasise something e.g. They couldn't have been more kind.

Kinder! I want to become a kinder person. Given the choice between two people, I'd prefer to spend time with the kinder person.

Typically with comparisons, if the adjective in English is just one syllable, you add -er to the end, but if the adjective is more than one syllable, it requires "more" before it. Examples: She is taller than he is (or smarter, or stronger, or fatter, etc). but She is more beautiful (or more clever, more industrious, more sophisticated) than he. An exception: you can't say "funner" even though "fun" is one syllable. You should say (as an example: Studying languages is more fun that watching TV)