How can I kill a tree stump that keeps sprouting? I want to stop it s roots from spreading.?

I'd say dig a hole there and fill it with different soil and put the dirt you dug up somewhere in the woods. Pouring chemicals into the ground is never good.

First, immediately cut off all sprouts. When young they will break off easily. Pile a little dirt on the cut surface and keep it moist to encourage rot. You can drill holes in it if you want to and fill those with dirt.

At a garden center you can get commercial products to encourage rot but I just used dirt and water. But be sure to keep the sprouts cut off.

Really would help if you had told us just what kind of tree you had.
There are a few trees that are difficult to kill... Like... Popcorn tree, and chinese sumac (ailanthus altissima).
But... Most trees are going to be easily killed... What I do... Is allow. The sprouts to grow for a year, and then hit them with an ax. They rarely send out new sprouts... By timing it right... It only requires a single effort... Trying all this hole drilling and whatever is a lot of unnecessary effort.

Drill some holes in the stump then using a cheap paintbrush, paint brush killer over the entire surface of the stump. You can also paint the leaves of any shoots. The brush killer will kill all of the roots preventing the stump from resprouting and seems to be harmless to surrounding plants. I have used this technique many times and it works very well, even on tough stumps like privet.

You can buy brush killer at pretty much any home center or garden center. I use the Bayer brand.
http://www.homedepot.com/p/Bayer-Advanced-32-oz-Brush-Killer-Plus-Quart-Concentrate-704640B/203124091

Buy a quart of Round Up concentrate, make a fresh cut on the stump and then use a dollar store paint brush to paint the round up onto the stump and or branches( with fresh cuts). This works because the cambium( food highway) is still active and will take the round up down to the roots and kill the plant.

They sell Stump Remover at any garden shop. I have also heard, that if you nail copper nails into it , it will die. I haven't tried that method and like I said it is only hearsay.

I am currently experiencing that very same thing. I'm not sure of the type of tree that this was, but it had several shoots the largest being two inches thick. These were cut down to approximately three inches above ground and holes have been drilled into each of these stumps and White Vinegar has been poured onto each stump, and into each hole. The effect has been that the new little shoots have stopped coming up and the above ground stumps appear to have shrunken and hardened, but I will need more time to see if this will finally kill the roots. I've been told by the employees at the nursery that I'll need to continue doing this for a few months to finally end the roots, but I'd rather do this than pour harsh chemicals into the soil.

Pour some undiluted white vinegar into the holes of the stump.

Drill a few holes in the stump and fill them with household bleach.

Use a torch, and burn as much of it as you can. If the sprouts are from the roots, you need to remove the roots. Just do a little bit each day.