Before ask the general question, let us check the motivating example.
Consider two transcendental elements (and they are algebraically independent) over $\mathbb C$, say $x$ and $y$. Then $\mathbb C[x,y]$ has transcendence degree 2 over $\mathbb C$. Now take subrings. One thing is
$$\mathbb C[x,y^2-x]$$
and another one is
$$\mathbb C[x+y,(x+y)^2].$$
The first one has transcendence degree 2 over $\mathbb C$, but the second one has degree 1 obviously.
So in general setting, consider a subring $A=k[y_1,\ldots,y_d]$ in $k[x_1,\ldots,k_d]$. Thus $y_i$ for $1 \leq i \leq d$ is a polynomial over $k$ with variables $x_1,\ldots,x_d$. How we determine that $A$ has transcendence degree $d$, or $<d$.
Best Answer
Let $B = k[x_1,\ldots,x_d]$ a polynomial ring and $y_i \in B$ for $i=1,\ldots,d$ and $A=k[y_1,\ldots,y_d]$. Consider the exact sequence
$$0 \to I \to k[W_1,\ldots,W_d] = C \xrightarrow{\phi} B$$
with $\phi(W_\nu) = y_\nu$. The ideal $I$ can be easily computed in a concrete example e.g. with Macaulay2. This is nothing more but a suitable gröbner base calculation. Iff $I=0$ then $\mathrm{trdeg}_k A = d$.
More generally $\mathrm{trdeg}_k A = \dim A = \dim C/I$ as $A = \phi(C) \cong C/I$ and $\dim C/I$ can be computed with Macaulay2 too - in general it can be read off from the leading terms of a gröbner base of $I$.