Let $K$ be a global function field, that is, a finite extension of $\mathbb{F}_q(t)$, where $\mathbb{F}_q$ is the finite field with $q$ elements.
What are the infinite places of $K$?
I know that $\mathbb{F}_q(t)$ has precisely one infinite place, namely that which corresponds to the change of variable $t\mapsto 1/t$. Furthermore, if $K/\mathbb{F}_q(t)$ is separable, then we can describe the infinite places of $K$ via the factorisation of the separable polynomial generating the extension, as in Corollary 13.6.
What happens when $K/\mathbb{F}_q(t)$ is not separable?
There are no infinite places in function fields, you mean places above $t=\infty$ ie. above $u=t^{-1}=0$.
$K = F_q(u)(c_1,\ldots,c_n)$, if $g\in F_q(u)[x]$ is irreducible and non-separable then $g' = 0$ so $g = G(x^p)$,
Let $p^{e_j}$ be $c_j$'s multiplicity in its minimal polynomial, then $c_j^{p^{e_j}}$ is separable over $F_q(u)$.
Thus with $E = F_q(u)(c_1^{p^{e_1}},\ldots,c_n^{p^{e_n}})$ we have $K/E/F_q(u)$ with $K/E$ purely inseparable and $E/F_q(u)$ separable, with the primitive element theorem $E = F_q(u)(a)$, let $d$ be the leading coefficient of $a$'s minimal polynomial $\in F_q[u][x]$, then $E = F_q(u)(da)$ where $da$'s minimal polynomial is $f(u,x)\in F_q[u][x]_{monic}$.
Since $K^{p^{[K:E]}}\subset E$, the places of $K$ above $u=0$ correspond to the places of $E$ above $u=0$ which correspond to the irreducible factors $f_i(x)$ of $f(0,x)\in F_q[x]$, each of them giving a maximal ideal $\mathfrak{m}_i=(u,f_i(x))$ of $R=F_q[u][x]/(f(u,x))$ which gives a principal maximal ideal $(\pi_i)$ of $R_i=(R-\mathfrak{m}_i)^{-1} R$ so that $v_i(y) = k$ if $y\in (\pi_i^k),\not \in (\pi_i^{k+1})$ is a discrete valuation on $R_i$ which gives a discrete valuation on $E$ and $v_i(y) = p^{-[K:E]}v(y^{p^{[K:E]}})$ is our discrete valuation on $K$.