If I have shown that $\text{GL}(n,\Bbb C)$ is a linear algebraic group, in particular, that the multiplication and the inverse are morphisms of varieties, then for any abstract subgroup $S$ of $\text{GL}(n,\Bbb C)$ that is a variety, can I obtain that the multiplication $\mu:S\times S\to S$ and inverse $\iota:S\to S$, such that this is just matrix multiplication and matrix inversion - via the restriction of my already shown morphisms of varieties above?
Say $\text{SL}(2,\Bbb C)\subset \text{GL}(2,\Bbb C)$,
Where we have $\text{GL}(2,\Bbb C)\subset \Bbb A^5$ from $(ad-bc)q-1=0$ for $\Bbb{C}[a,b,c,d,q]$ where $q$ plays the role of $\text{det}^{-1}$.
We had $\mu:\text{GL}(2,\Bbb C)\times \text{GL}(n,\Bbb C)\to \text{GL}(n,\Bbb C)$ via:
$$\mu(a,b,c,d,q,a',b',c',d',q')=(\phi_1(x),\cdots,\phi_5(x))$$
Where our notational choice corresponds to: $$\begin{bmatrix}\phi_1(x)&\phi_2(x)\\\phi_3(x)&\phi_4(x)\end{bmatrix},$$ and $\phi_5(x)=qq'=\text{det}(M_1)^{-1}\text{det}(M_2)^{-1}$
So to work with $\text{SL}(2,\Bbb C)$ we just have $q=q'=1$, so since $\phi_i$ are in $\Bbb{C}[a,b,c,d,q,a',b',c',d',q']$ surely restricting to $\text{SL}(2,\Bbb C)$ these are polynomial in $\Bbb{C}[a,b,c,d,a',b',c',d']$ and the multiplication is a morphism of affine varieties?
I think you need to be careful with the statement "subgroup which is a variety": you need the variety structure to be compatible with that of $\mathrm{GL}(n, \mathbb{C})$, in the sense that it defines a subvariety of $\mathrm{GL}(n, \mathbb{C})$.
Take for example the group $H$ of matrices of the form $$\begin{bmatrix} e^x & 0 & 0 \\ 0 & 1 & x \\ 0 & 0 & 1 \end{bmatrix}$$ $H$ is isomorphic to the additive group $\mathbb{A}^1_\mathbb{C}$, and so it does have a variety structure. However, its obvious embedding into $\mathrm{GL}(3, \mathbb{C})$ is not polynomial, and will not define a subvariety.
However, I think this is the only thing that can go wrong: the multiplication and inversion maps will work as set maps for any subgroup of the group, you just have to make sure that subgroup is a subvariety.