I've both had an Alchemist in the party and played one and have seen 2 very different builds:
I personally played a Bomb-Focused Mindchemist who popped Cognatogens and threw bombs for higher DC & damage. Decent overall, and very good Battlefield Control.
The player ran a Vivisectionist Beastmorph Alchemist focused on Mutagens and simply became a hit-and-run melee expert, using Pounce on a flanked enemy. He was easier to kill than a gnat, but hit like a runaway freight train.
Now, since your player wants to be melee-focused, I suggest the latter over the former. The Beastmorph archetype gives your mutagens the Beast Shape spell line, which gives you pounce. Later on, you can get flying, but really, if you are going pure melee, pounce is a godsend for "nova fighters" like the melee alchemist builds.
My 2c on this subject, and I hope that the player can see the value of this.