"Because what you can control... can't hurt you."
The philosophy of the Dominion was divided into three distinct groups, each with notably differing outlooks and aspects. The perspective of the Founders, or the Changelings, with whom ultimately rested the control over the Dominion, was formed from their history of persecution at the hands of non-shapeshifting lifeforms they thereafter termed "Solids". To that end, the priority of the Founders was the survival of their own species, by any means necessary. They had no interest in matters such as Klingon honor, the Federation's goal of peaceful exploration, Ferengi material success, or objections made by opposing groups regarding their methods of self-preservation. (DS9: "The Search, Part II") Their philosophy is to dominate everything that can be dominated and destroying all that cannot; the Founders were, in essence, driven by an urge to "impose order on a chaotic universe."
The Founders' extreme longevity (indeed, practical immortality) has provided them with a uniquely long viewpoint. As the genetically-engineered and highly intelligent Jack described: the Dominion does not adjust its strategies based on what has occurred within the past week or even year, but is concerned instead with what the universe will look like centuries or more forward. This perspective was evident in the Dominion War, where the strategy was to engage in a long-term war of attrition, counting on superior construction methods and their ability to breed Jem'Hadar, rather than risk everything on one battle. (DS9: "Statistical Probabilities")
Apart from the Changelings' metamorphic abilities, the most distinguishing ability of their species is "linking" – the physical and mental connection of two or more Changelings. The species seems to exist in a collective union called The Great Link for much of their life span, producing a strongly anti-individualist perspective. Consequently, the Founders seem to be a remarkably unified, even monolithic, species. Their most sacred axiom: "No Changeling has ever harmed another" reflects both this and their obsession with physical security. Disagreement between the Founders, however, is not unheard of, as illustrated by the (at least initial) lack of consensus over how to deal with Odo after he caused the death of another changeling. (DS9: "Broken Link")
The Changelings genetically-modified the Vorta to serve them in various roles. They have also genetically-engineered the Jem'Hadar to serve as their soldiers. Each group of Jem'Hadar is closely controlled by the Vorta. The Jem'Hadar enforce the will of the Founders, fight in wars to expand the Dominion, protect the Vorta and the Founders, etc. Both races are engineered to worship the Founders as their gods.