Of course what appears to be a rambling, ad hoc act was nothing of the sort. The wide eyed grimaces, the failed tricks, the funny walks were micro rehearsed. That takes an enormous effort and as with many other great comics, something has to give. Cooper drank and played hard.
In a play of two halves, we see the public Tommy and then a version of the private one. Clive Mantle is terrific as Cooper, capturing his essence in an extremely convincing impression.
As for being the funniest man in the world, there are certainly moments in the first half, where you really believe the claim.