To clean a tungsten ring, it is very simple.  You can use any jewelry cleaner you have and soak it for 5-10 minutes.  When it comes out, just wipe it off with a paper towel.

If you do not have jewelry cleaner laying around, just use Windex.  Take the ring and while over a sink, spray the ring on the inside and outside.  Soak it. 

With your thumb, rub around the ring until you hear it start to squeak.  Once you hear that squeak, you know it is clean.

Run the ring under warm water and dry it off with a paper towel.

Your tungsten ring should look brand new again!