Covering an area with black plastic or membrane will kill grass and many other weeds. Bramble may be strong enough to break through.
Even if you kill the bramble using chemicals, you'll still need to dig out the dead roots before you can cultivate. You might as well get the roots out now. My preferred tool for removing brambles is a mattock. It's actually quite easy. They are not deep.
Also, a membrane will not kill weed seeds that are dormant in the soil. A covering prevents them from receiving light, which is necessary for germination. Weed seeds can remain dormant in the soil, and viable, for many years. As soon as you remove the membrane, to start cultivation, they'll know it's time to wake up. You'll simply need to keep weeding, for some years.