Here is an approach that uses disjunctive elimination. Given the following I assume the DeMorgan rules are available:
I can expand out ~(C & B) into ~C OR ~B
On line 3, I used DeMorgan rules (DeM) to get a disjunction which I will then need to eliminate to reach the goal.
To eliminate the disjunction, I have to consider both disjuncts, "¬C" and "¬B".
The first disjunct is the easiest, but it may be confusing because it is so easy. The assumption, "¬C", for that subproof is precisely what I want to show. There is nothing more to do (at least for this proof checker).
The second disjunct uses explosion based on the observation that lines 1 and 5 are contradictory. This proof checker allows me to state the contradiction (⊥) on line 6 and use explosion (X) on line 7 to reach the conclusion that I want "¬C". The one you are using may require something different.
Since I have the same conclusion for each disjunct I can discharge the two assumptions on lines 4 and 5 by citing the rule of disjunction elimination (∨E). In this proof checker I have to reference the disjunction itself (3), the first subproof (which is only line 4 but written as a range 4-4), and the second subproof (5-7).
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/