I have combination skin that is not overly oily on the T-zone, neither is too dry, but I do have some blackheads. I use Avene Cleanance Soapless gel cleanser, it removes all makeup, even eye makeup really well, leaves my face squeaky clean. It can dry out your lids if you use it on your eyes everyday, though (speaking from experience).
For my eyes, I use a product that can also be used all over the face and is great, it's the Vichy Purete Thermale 3 in 1 makeup remover (cleanser, toner and eye makeup remover in one). It's really nice too.