Does this sentence use correct grammar You should of known better.?

No

The sentence "You should of known better." derives from "You should've known better."; "should of" sounds about like "should've". Of course "should've" is a contraction for "should have" so the correct sentence is either "You should have known better." or (if speaking informally and using contractions) "You should've known better." Writing the sentence as "You should of known better." instead of "You should'veknown better." is INCORRECT.