No. The Klingon empire is a totallitarianist empire. Sort of like a closed union shop. You are either in or you are OUT! All the way out. Know what I mean?
The Klingon Empire isn't like the Federation, where many different species coexist, but the Klingons have been known to annex planets, making the inhabitants subjects of the Klingon Empire. A good example of this is the Enterprise episode, Judgement.
The Klingons do not expand peacefully like the Federation, instead they prefer to conquer. As mentioned, the aliens from "Judgement". They also conquered Krios. (mentioned a few times in ENT and TNG) We don't know the standing of those people within the Empire, but they are likely regarded as second-class citizens at best. However, judging my Klingon culture, it would seem that it might be possible for a non-Klingon to raise his stature within the Empire by acting honorably or otherwise via marriage (DS9 "The House of Quark", DS9 "You Are Cordially Invited") In those instances, Quark briefly had is own Klingon Great House, and Jadzia officially joined the House of Martok. Basically, if you can impress a Klingon in a position to allow you into their House, a non-Klingon CAN become a citizen of the Empire.