As you said that you know in your comment, the 3rd sentence is correct. 1 and 2 are not.
2 is no good because you repeat the word “like” without repeating the subject, and the punctuation doesn’t work. The following would be grammatical:
I like sports; I especially like playing badminton.
The semicolon separates the two related but different points. This could also work as two different sentences.
1 doesn’t work too well, also because of missing punctuation. And the word “playing” seems out of place- the way it’s currently worded, it sounds like “playing badminton” is a sport you especially like, which doesn’t make sense. You could either say badminton is a sport you like, or badminton is a sport you like playing. So either of these:
I like sports, especially badminton.
I like playing sports, especially badminton.