You can kill the roots with chemicals - but they will still be there, I am afraid to get rid its either digging them out or if its tree roots a stump grinder which can be very expensive.

If you are talking about fine or matted roots, it might take a few seasons of patient digging and removal as you go. I would be impatient to plant things so would clear a patch big enough for a small group of plants and gradually get round the garden like that. Perennial weeds will pop up over and over again but will weaken if you keep at it! (I'm having an ongoing battle with bindweed and running strawberry and sometimes I think I'm winning and other times I despair!)