When I make a cup of coffee by mixing coffee and hot water to make a solution is there a process where I can get the coffee back? How does it work?

Yes. You are describing the process of preparing a cup of instant coffee. When you mix water and instant coffee, a physical change occurs. The coffee solids are dissolved and suspended in the water. Physical changes can often be reversed, while many chemical changes cannot be reversed. 

You have two options for recovering the coffee. You could filter the coffee solids out of the water or you could allow the water to evaporate, leaving the coffee solids behind. The coffee particles are very small. You would likely have a hard time finding a filter fine enough to remove everything. Gently heating the solution to evaporate the water will likely be more effective. You should be careful not to heat the coffee too much, as excess heat might cook the coffee. Cooking is a chemical change that cannot be undone.