Does coffee allow you to use a higher percentage of your brain?

Of course not. Caffeine is a drug that have just the ability to prevent your brain to start the sleep process. You will still feel tired and need sleep, but you will be able to maitain some focus and stay awake as long as caffeine is active in your bloodstream. It is similar to ecstasy, that causes similar effects in this respect.
For example, if you need to stay awake to do some physical exercise, you can drink coffee and stay awake and do the exercises, but when the effects turn off, you will feel tired and exausted. Caffeine just “block” the signals in your body that signalizes for you to rest and sleep, nothing much more than that. It has important e useful effects in the respiratory and cardiovascular system as well.

There is a myth in this question, a popular one, in that we use a limited percentage of our brain.Truth is far more complex than this, for a start brain tissue is metabolically costly, what isn’t needed isn’t kept. Muscles atrophy when unused for the same reason.

What percentage of the brain is used for doing what task is not clear either so the question of any drug increasing the usage for a given task is impossible to answer.

Finally as the brain gets more efficient at a task it need to use less resources to do it. Therefore paradoxically you don’t want to have an increased percentage use even if you could pin this down. If the organism could walk using one neuron only, it probably would. Our biology would tend to the more efficient usage scenario – though it may be a false valley of efficiency if measured on a graph at times – hence why there are many evolved solutions to biological concerns like motion.