is it a particular form of かっこいい? Yes, that's right. It is called katsuyo, 活用. Conjugation. See the verb なる（なり, in your example). When it takes かっこよい, the い changes to く, and so かっこよく. かっこよい＋なる -->　かっこよくなる You ask any grammatical functions? It is just conjugation. You know なり has got the conjugation, too. かっこよい is in a dictionary form. かっこいい and かっこよい are both OK, and かっこいい is more casual. For the meaning of かっこいい, try jisho org. https://jisho.org/search/%E3%81%8B%E3%81%A3%E3%81%93%E3%82%88%E3%81%8F

glad to be helpful. just like what u said, yes the word かっこよく is a shape-change of jap. so lets begin with that. かっこいい(the word いい can also written in kanji 良い) just one step to かっこよく(て)

Thanks a lot