.005 isn't that far out for a 3 jaw chuck, especially an older one. Even good 3 jaw chucks will run .002-003 out. If you want closer tolerances than that, find a 4 jaw. They're a PITA to setup, but once you get good, you can get parts running true to .0001 in a minute or less. Another option is a Buck (or buck style) 3 jaw Adjust-Tru type chuck.
You should also check it at several diameters since runout can vary quite a bit. If the chuck was used most of the time with 1" stock, you'll probably find more wear and more runout there than you will if you chuck a 2" bar.
Disassembling and cleaning it is definitely a good idea.
In theory, whenever you mate a chuck to a machine, you should remove it from the backplate and take a skim cut on the backplate to ensure concentricity with the spindle.
Make sure that the spindle threads and chuck are immaculately clean, then clean them again. Run your finger across all the surfaces to feel for any debris or dings. Small dings can be stoned down. Also make sure that the backplate threads all the way on and that the register is mating right. The register aligns the chuck, NOT the threads.