Climbers that attach themselves by means of adhesive tendrils and the ones that are self layering are extremely effective in suppressing weeds when they are grown as ground covers. You can also grow twining climbers as ground covers as long as you make sure that the shoots are spread evenly across the ground. If needed you can use wire hooks to keep the shoots in place.
When growing climbers as ground cover you must take care of the plants that exist in the area where you are growing them because your climbers might use them as support when they reach them. It is best in this case to use less vigorous climbers or to move the existing plants in another location to avoid them of being swamped by the climbers. This way your climber can spread freely.
Here is a short list of climbers that you can use successfully as ground covers: Asteranthera, Clematis, Hedera, Kennedia, Parthenocissus, Pileostegia, Plumbago auriculata, Tecomaria, Trachelospermum, Tropaeolum speciosum.