dark circles under the eyes are hereditary, so you can't completely get rid of them. what you see is blood in the vessels because the skin there is so thin.
what does seem to help is raising the head of your bed a few inches, so that, while you are sleeping, the blood doesn't pool under your eyes. taking vitamin K supplements (strengthens blood vessels) is also of some benefit.
dark circles may be caused by different things. It may be hereditary, liver related, heart related, deep sleep related, stress, diet, you name it. It is best to find out what is causing his dark circles first and then seek action.
In the mean time, to cover up, the best concealer I ever tried is the one from Bobbi Brown. Creamy concealer kit. It's perfect
2C-3A. PW: on request
Love:Jessicurl WDT+Too Shea, Coconut Oil, Aveda Rosemary Mint shampoo, BRHG and Curls Rock Creme.
Mine are hereditary, and the only things that have ever helped are drinking bucketfuls of water, cutting out all caffeine and sodas, and undereye concealer. The Bobbi Brown one is great, and I've also had great success with Stila's. Tiny little pot that lasts forever.