What this "he could chalk his billiard-cue with his knuckles" indicates?

Gout is a medical condition, which, in its chronic stage, can give crystalline deposits of a chemical called uric acid in various joints of the body. These deposits are called tophi (which is why this form of the condition is called "tophaceous gout"). They often have the appearance of small fragments of chalk just underneath the skin of the joints.

Here is an image showing these deposits:

The chalky appearance is actually clearer in the image of the toes.

Now note that chalk (actual chalk, which is the mineral calcite, or calcium carbonate) is often rubbed across the end of a billiard cue to enhance friction to increase accuracy of the shot. The passage is fancifully comparing the chalky appearance of the poor old chap's gouty knuckle joints to the actual chalk used in the game of billiards and imagining that he could use his knuckles to directly "chalk" the cue stick (clearly this can't literally be true).

The second guess is almost right. Gout makes your joints crumbly, so they could be used as a chalk for the billiard-cue, not his actual knuckles.