As far as I know, a paper of Alon and Frankl "The Maximum Number of Disjoint Pairs in a Family of Subsets" (available here) contains state of the art knowledge on the problem.

Briefly outlining some of its conclusions, let me mention that for reasonably large $k$ keeping the sets in $C$ supported on disjoint subsets of $[n]=\{1,2,\ldots,n\}$ works much better than keeping them individually small. For example, for even $n$ and $k=2^{n/2 +1}-1$, if we choose $C$ to consist of sets of smallest possible size than typical set in $C$ will have size $\Omega(n/\log{n})$ and two such sets almost surely intersect. On the other hand, if we choose $A$ to be the set of all subsets of $\{1,\ldots,n/2\}$, $B$ to be the set of all subsets $\{n/2+1, \ldots, n\}$ and $C =A \cup B$, then at least half of the pairs of sets in $C$ are disjoint. Solving a problem of Erdős, Alon and Frankl show that this example is essentially the best possible.

As far as I know, a paper of Alon and Frankl "The Maximum Number of Disjoint Pairs in a Family of Subsets" (available here) contains state of the art knowledge on the problem.

Briefly outlining some of its conclusions, let me mention that for a wide range of values of $k$ keeping the sets in $C$ supported on disjoint subsets of $[n]=\{1,2,\ldots,n\}$ works much better than keeping them individually small. For example, for even $n$ and $k=2^{n/2 +1}-1$, if we choose $C$ to consist of sets of smallest possible size than typical set in $C$ will have size $\Omega(n/\log{n})$ and two such sets almost surely intersect. On the other hand, if we choose $A$ to be the set of all subsets of $\{1,\ldots,n/2\}$, $B$ to be the set of all subsets $\{n/2+1, \ldots, n\}$ and $C =A \cup B$, then at least half of the pairs of sets in $C$ are disjoint. Solving a problem of Erdős, Alon and Frankl show that this example is essentially the best possible.