Hang,
Refreshing old checkering is fairly easy, the layout (hardest part of the job) is already done for you. The difficult part is getting the correct tools. When I did it to a couple of shotguns, I used a thread gauge to determine spacing and then bought the appropriate cutters. You will need a single line cutter, a long single line cutter (to straighten a crooked line), and then you have options. A cutter with a blind vee and one or two cutters will speed things up but it will also produce more mistakes faster. I like the single cutter with the guide (blind vee). Get a couple of each cutter and two handles. Set up one handle to cut on the push stroke and one to cut on the pull. It will help speed things up.
Once you have all your tools, it is the same as tackling a blank stock. Your layout is done so start with the two master lines, one each way. Then go out from there. Borders are cut last to clean up any over runs.
Refinish the stock first, if you are going to. No sense ruining the work you put into cutting that checkering.
(Edited to ad). Cut your master lines full depth but when you start to add in the field cut them 1/2 depth, do all of them in one direction, then cross the, 1/2 depth. If you try to cut them full depth the first pass, you will lose the cross pattern. Two or even three passes will give a nicer result.
Good luck! Keep us posted.
Kevin