They're both wrong, I'm afraid. The correct expression is "That is a way FOR ME to improve my English skills..'. As for the pronoun, remember that after a preposition - to, for, with, by and so on - you always need an object pronoun. The object pronouns are me, you, him, her, it, us and them.

I would put it as "That is a way for me to improve my english skills because I will..."

"That is a way for me to improve my English skills." Is a correct sentence, but is lacking in detail. If you start out the sentence with "That", well, what is "That"? is it a specific method of studying, or a trick that you know of? The word "that" needs a little more description for it to make sense in a conversation. You could start of the sentence with "There". So, "There is a way for me to improve my English skills, yadda yadda yadda..." and then you could go on to detail what the actual "way to improve my English" is. If this is confusing at all, lemme know and I'll try explaining it differently! Good luck brah.