If $X$ is a simplicial set which is not Kan, you can compute the homotopy groups of $X$ by choosing a weak homotopy equivalence $f: X \rightarrow Y$ where $Y$ is Kan and then applying the construction you are familiar with to $Y$. There are many ways of characterizing the relationship between $X$ and $Y$ without every mentioning topology or model categories (though I'm not sure it is so helpful to avoid these). For example, you can take any map $f: X \rightarrow Y$ with the following property: for every Kan complex $Z$, composition with $f$ induces a bijection from $[Y,Z]$ to $[X,Z]$, where $[K,Z]$ denotes the set of maps from $K$ into $Z$ up to (simplicial) homotopy. There are several purely combinatorial constructions of $Y$ from $X$: for example, Kan's $Ex^{\infty}$ functor.

If $X$ is a simplicial set which is not Kan, you can compute the homotopy groups of $X$ by choosing a weak homotopy equivalence $f: X \rightarrow Y$ where $Y$ is Kan and then applying the construction you are familiar with to $Y$. There are many ways of characterizing the relationship between $X$ and $Y$ without even mentioning topology or model categories (though I'm not sure it is so helpful to avoid these). For example, you can take any map $f: X \rightarrow Y$ with the following property: for every Kan complex $Z$, composition with $f$ induces a bijection from $[Y,Z]$ to $[X,Z]$, where $[K,Z]$ denotes the set of maps from $K$ into $Z$ up to (simplicial) homotopy. There are several purely combinatorial constructions of $Y$ from $X$: for example, Kan's $\operatorname{Ex}^{\infty}$ functor.