You can either use a glyphosate based weed killer or a triclopyr based weed killer.
Both will do the job well although the Glyphosate based one may need an additional application after a few weeks.
The main difference between them is that triclopyr won't kill grass and it won't allow other plants to grow on the treated area for up to 6 months. Glyphosate will kill anything it touches but it's safe to replant shortly afterwards and it leaves no residues as it breaks down in the soil.
What is the best weed killer for brambles?
That depends entirely where your brambles are growing. If they are growing in grass or in an area of the garden where you don't plan on planting then use the triclopyr based SBK brushwood killer. If you want to plant in the area after you've killed the brambles and it's not grassy then use glyphosate based weed killer. Both will kill your brambles and stop them from growing back.
Whichever weed killer you use you may still have to pull out or dig out your bramble bush. If it's large and many years old it will take a long time for it to break down naturally. After 3-4 weeks you can dig them out and dispose of them via your recycling centre.
I think the best way, is cut them back with loppers,with rake off the area and dig out the roots with a fork or mattock.
Heavy duty gardening gloves will definitely be needed.
As per Flonska's answer....a strong Glyphosate mix usually does mine. Give it a week to get in the system then just cut the brambles back....the roots should effectively be dead. No back breaking digging etc.