The full quote appears to be "developed a rigorous theory of levers and the kinematics of the screw", and is repeated by many authors after History of Technology by Dimarogonas. The rigorous theory of levers is developed in Archimedes's only surviving mechanical work On the Equilibrium of Plane Figures, along with the law of buoyancy, but it is hard to say what Dimarogonas means by "kinematics of the screw". We know (from Pappus's Collection) of a classical work that analyzes screw motion as a composition of uniform linear and circular motions, About the Screw, but it is by Apollonius rather than Archimedes, although it was likely motivated in part by Archimedes's earlier work On Spirals. Its content is discussed in detail in Acerbi's Homeomeric Lines in Greek Mathematics.
But Archimedes's main contribution was creating a first mechanical theory, the theory of simple machines, which can be applied to the screw just as to the lever. It is best characterized not as kinematics, since it dealt with forces and weights, but as statics, or better quasi-statics, since motion was assumed to be of a kind where at each instant the moving body is at equilibrium to a good approximation. Here is assessment of available sources on this theory in Russo's Forgotten Revolution:"we can reconstruct some features of third century B.C. mechanics by combining information from three sources: the one surviving Archimedean book; documents, particularly works of military technology, that mention machines actually built; and treatises written centuries later, above all Pappus' Collection and Heron's works. Among these the most useful is the Mechanics, which describes the five simple machines (the winch, the lever, the pulley, the wedge/ramp and the screw) as well as a number of composite machines designed for various uses. The pseudo-Aristotelian Mechanics, which share many features with Heron's Mechanics, also has interesting information".
And here is his reconstruction of the theory itself:"The problem, given a maximum available force F and the need to lift a weight W, is to design a machine having the appropriate mechanical advantage [the ratio W/F] and configuration, so the weight can be lifted by applying the available force at a convenient point and in a convenient direction. All devices of this type can ultimately be traced back to the simplest such device, the lever, which Archimedes uses as the starting point of his scientific theory of mechanics. Of course problems of this type had always been around and had often been solved practically, as far back as paleolithic times... The qualitative leap made possible by science lay in that now one could compute the mechanical advantage theoretically, and so for the first time design a machine from first principles. This leap surely took place as early as the third century B.C. Pappus and Plutarch tell us that Archimedes had solved the problem of lifting a given weight with a given force; in other words, he knew how to design a machine with a specified mechanical advantage". As applied to the screw this would mean that given its length and height for example Archimedes could calculate the cross section and the handle size to make it possible for a person to operate it, and solve other such problems.