Solution 1:

The correct sentence is

How does he do that?

In the same way one writes

How does he run?

How does he fly?

Solution 2:

As @Jasper said, the answer is

How does he do that?

The first "does" is the auxiliary verb which allows to turn the sentence into a question. The second "do" is the verb of the action in itself, which is not conjugated because the conjugation is applied to the auxiliary verb.

The sentence is perfectly correct. If you feel awkward about how it sounds, you could say for example:

How can he do that?