well 2 common problems to take a look at first would be the handbrake actuator levers and the cable itself, they are the little levers the cable pulls on when you pull the handbrake.
The actuator levers are at the drum, what the cable pulls on.. To get them off you need to drop the drum off and unhook it from the shoes and feed it through the slot out the rear of the backing plate. They are renowned for seizing. You can just replace but usually if you spend some time cleaning you can free them up, lots of WD40 as they melts the rust as well and lubricating. you may need a hammer to free it up but once its freeing up keep turning and spraying WD on it. Eventually it should move super freely, whack some copper slip on it and put it back together.
As they are getting older it seems more and more common for the handbrake cable to seize or not move freely inside its sheath. Now I have never bothered trying to fix this, perhaps spraying silicone spray up it and wiggling for a long time would free it but its only gonna be a temp fix. But with it disconnected from the levers at the back and the handbrake handle check the cable slides nice and easily. If not just replace it with a new one is best course of action.
Other than that you are just checking the usual stuff really, weeping at any brake line connectings, pinholes in the hard-lines, the wheel cylinder rubber boots are good and not leaking fluid, that you don't have a big lip on the inside edge of the drum, there is friction material left on the shoes and stuff like..