The terms "converse" and "contrapositive" apply only to statements of the form "If A Then B". In that case the "converse" is "If B Then A" and the "contrapositive" is "If NOT B then NOT A".
But you do NOT have an implication here because you have no "then".
You probably mean "if a> b an b> c then a> c". (Which is true.)
The "converse" is "if b> c then a>b a and b>c" (Which is not true.) The "only if" formulation is equivalent to the "if then" formulation but "if X then Y" is equivalent to "X only if Y". You have "X" and "Y" reversed.
The contrapositive is "if NOT (a> c) then NOT (a> b and b> c)". We can reduced that a little by using the fact that "NOT (X and Y)" is the same as "NOT X or NOT Y" as well as the fact that "NOT x> y" is the same as " " to write the contrapositive as "if then either or ". (which is true. The contrapositive of a statement is true if and only if the statement is true.)
You seem to think that "NOT x> y" is the same as "x< y" (y> x) but that is not true. If x= y then "NOT x> y".