Chemical composition is just the start point, this determines basic properties at microscopic scale based on atomic link forces, solid solution, precipitation and crystalline structure (ordered arrange of the atoms) by say someone.
Crystalline structure give us the first approximation to macroscopic scale behavior of the material, each type of unitary cell in a crystal have different properties based on the distance between atoms, slip systems, presence and type of solutes (alloy elements and even impurities), by example: Fe alpha (ferrite) crystalizes in BCC system, during heating in a heat treatment, turns in Fe gamma (austenite) and FCC system, this allow that more solute (carbon) can ingress to the crystals of austenite, but whit the accelerated cooling during quenching, all of the solute can't exit from the net promoting the crystallization in a different system, other than ferrite BCC, named Martensite whit BCT system. The stress generated by this forced arrange gives high hardness, low ductility, low toughness to the material, thus the quenched material could be tempered for relax some stresses and obtain the desired properties.
Next step to analyze is polycrystalline materials (most of the utilized in engineering). Be a crystal the minimal unity whit the same atomic arrange and named "grain", in polycrystalline materials, each grain are surrounded by others one, hence grain boundaries exists. Each grain also has imperfections like dislocations. The dislocations as grain boundaries obstacles the slip systems of the crystal, interacts whit crack propagation and defines in part the yield strength of the material.
Grouping and displacement of dislocations (or even others defects) could promote hardness, more or less ductility and UTS by strain hardening, affecting the plasticity, toughness and maximal strength of the material, here chemical composition could play a role in a hardening mechanism by precipitation, the presence of precipitates formed by alloy elements affect also the deformations systems, but the precipitation shall be controlled, fine and homogenous, its regularly achieved by thermomechanical processing of the metal whit controlled rolling for example.
Next step: the texture (organization and orientation) of the microstructure.... mmmm, may be later. In essence, the manufacturing processes modify, orient and / or order the microstructure of the material to favor one or more properties.
The mechanisms are varied, the foundations are extensive and in this brief explanation only some of the many possible ones are mentioned.
I recommend reading introductory texts to the science and engineering of materials, Askeland and Phule has a good one to start.