evolution - How exactly are game theoretical evolutionary models described during implementation for computer simulations?

There is no single way to build such a model. They can vary from a simple mathematical statement like Hamilton's rule (rB>C) to the chemical diffusion models used to describe the patterns in animal skin coloring (like zebra stripes, leopard spots and the like).

There are efforts to build molecular models of entire cells like this model of mycobacterium genitalium dividing, which integrates nearly 30 different mathematical models to describe different aspects of the organism. There are efforts to build such a model of an entire brain as well.

Another common sort of model for evolutionary biology is the use of game theory, where different strategies can be posed one against another as in the prisoner's dilemma competition Dawkins describes in the Selfish Gene.

It goes on and on. Basically biological modeling is driven by the sorts of mathematical models that we know. New models will reveal new paradigms of how biology works. They can be highly mathematical, but their relative importance and when they apply and what they mean are more analogy than proof.

For instance in the prisoner's dilemma, the first contests showed that Tit for Tat was the strongest model - generally assisting others, but betraying when there is a history of betrayal. The ideas at the time moved towards general cooperation in populations. More recent replays have shown that if there is a team of entrants that make extraordinary gifts to each other (allow betrayal without retribution), then they can compete against other models quite well.

One can never prove that a selfish model for the prisoner's dilemma will not show up, though biological systems to seem to be highly cooperative. That is a model, not a proof.