Shimano Sis Bar-end Shifters on a Shimano Deore xt derailleur Rebuilding an original fisher MTB, with drop bars and bar end shifters. 

I was wondering if I am going to run into any compatibility problems if I run this with a Shimano Deore Xt derailleur system.

Also I am wondering what type of chain would be best to use?
It depends on which generation of XT derailer and how many speeds your cassette is.

For the last generation of 9spd and earlier (RD-M772 and lesser numbers), the actuation ratio is all the same for mountain and road. So you would use SL-BS77 shifters with a 9 speed cassette, SL-BS64 for 8, etc, and the shifters won't care which derailleur they're attached to. 

Shimano changed their MTB RD actuation ratio when they introduced 10-speed mountain components, and this is where the answer gets more complex. If you're using any 10-speed Shimano RD, the only choice I know of for bar end shifters (unless you use a pull adapter) are the ones made by Microshift specifically for this purpose. Road/tri bar ends such as SL-BS78 will not work because they're for the older actuation ratio. And if you're using a 10-speed XT derailer, you must use a 10-speed cassette.

Same deal for 11. If you have the 11-speed XT derailer RD-M8000, the only bar end shifter choice (again that I know of) is the one made by Microshift.

Use whatever chain your cassette is made to take.