Based on Qiaochu's answer, we can explicitly work out the axioms, and hopefully reduce the messiness somewhat:

The empty set is compact.

Any intersection of compact sets is compact.

A finite union of compact sets is compact. (These replace the axioms of a topological space)

If the empty set is the intersection of infinitely many compact sets, then it is the intersection of a finite subcollection of them. (This says that our compact sets are actually-compact-in-the-new-topology.)

For any point $P$, there are compact sets $A$ containing $P$, $B$ not containing $P$, such that for any compact set $C$, $C- (A \cap C ) \cup B$ is compact. (This replaces local compactness, with $A - (A \cap B)$ the open neighborhood)

For any pair of points $P_1,P_2$, we can find $A_1,A_2,B_1,B_2$ satisfying the conditions of the previous axiom such that $A_1 \cap A_2 \subset B_1 \cup B_2$. (This replaces Hausdorff.)

We do not need a condition to prove that all the sets that are-actually-compact-in-the-new-topology are compact, because the local compactness axiom gives us an open cover such that anything covered by finitely many open sets is covered by finitely many compact sets, so if it's closed, it's compact.

It's possible that some of these axioms can be simplified further.