Prenote: in America, we generally use the mouth to denote extract other people's emotion, but in Japan and other oriental nations, they generally look at the eyes because it is harder to fake. Thus comes the saying, "the eye is the window to the soul."
That being said, I believe Hellsing takes it one step further, that being that the eyes are the soul.
Vampires are said to possess a third eye that lets them have near out of body experiences when using it, and I believe the difference between a standard vampire and Alucard is the manner in which they feed.
For most vampires, the only soul they will accept into their being is their own, therefore the only eye they gain is the third one. This causes Alucard's disdain for them not being a true Vampire. However, Alucard accepts the souls he feeds on into his being allowing for the growth of additional eyes and additional power.
Therefore, the eyes we see in the show are just some of the people he has consumed and accepted.
That's just my opinion anyways.