With great difficulty. Usually the approach is to suppose that you have an algorithm and show that either (a) this algorithm allows you to do something impossible (e.g., solve a different impossible problem, or reach a contradiction); or (b) there are two different problems with different answers which the specified algorithm cannot distinguish between (this is generally only possible where you're proving that it's impossible to compute something in less than some number of steps).

The first approach is used for things like showing that the halting problem is impossible; the second approach is used for things like showing that it's impossible to have a comparison sort which runs in less than O(N log N) time.