There are two basic ways to do this: you could use a bitmap or you could draw it manually.
The bitmap solution is probably going to produce a better looking result (as well as being easier to "skin") and annoyingly, it's probably easier to do as well.
Since this is backgammon, there are only 30 pieces, so it wouldn't be a major problem to draw your counters via a PictureBox (rather than drawing the image manually in the paint event. All you have to do then is move them by changing the Position property.
Google should be able to help you find a copyright free image of a board that you like - then just set it as the background image for your form or control.
This is the easy bit - making it a good player, now that's going to be difficult!