How do you upholster the back of dining room chair?

With no picture it is hard to say.

Some chairs have material on both sides.
Look for any screws that are holding the back to the chair and remove them.
If there is material on both sides then there is probably staples on the bottom, remove the staples and carefully remove the old material. It will either be stapled or sewn on the sides. If it is stapled remove them, if it is sewn slide it off the frame.
Once you have removed the old material lay it flat, if it is sewn you will have to use a seam ripper and open the sides.
Use the old material for a template to cut the new piece.

Once that is done just reverse what you did previously. Sew or restaple the sides and then staple the bottom then replace in the chair and put the screws back in.

If the chair has material on only one side, remove the screws and it will probably be stapled on the back.
Remove the staples, cut a new piece of material and restaple. Start in the center of each side working out to the corners and do the corners last. Just pull the material tight to get a nice fit.

Good Luck, if you can post a picture.