Since the question has arisen whether the standard arguments really do work with classes, let me post this answer giving some fuller details about one comparatively concrete way to do it. Other methods are also possible.
I shall work in the theory of Goedel-Bernays GB set theory (without global choice), a general setting for treating classes, which forms a conservative extension of ZF. This theory includes as a special case the traditional treatment of classes as definable collections in ZF, since every model of ZF, when augmented with its definable classes, forms a model of GB. Thus, the GB context seems the most comprehensive way to answer (and if you prefer ZF, then just imagine that all classes here are definable from parameters).
Suppose that $A$ and $B$ are classes and that we have class functions $F:A\to B$ and $G:B\to A$ which are injective, as in the question. Let $A_0=A-\ran(G)$, which is a class, since it is definable from $A$ and $G$. Let us say that a sequence $\langle x_0,x_1,x_2,\ldots\rangle$ is a back-and-forth-iteration sequence if $x_0\in A$ and $x_{2n+1}=F(x_{2n})$ and $x_{2n+2}=G(x_{2n+1})$ for all natural numbers $n$. Let $A_n$ be the elements $a_{2n}$ that appear at the even coordinates of a back-and-forth iteration sequence with $a_0\in A_0$. This notion is definable from $F$ and $G$. Let $A^+=\bigcup_n A_n$, which is a class definable from $F$ and $G$.
Let $H$ be the function $(F\upharpoonright A^+)\cup(G^{-1}\upharpoonright A-A^+)$. I claim that this is the desired bijection between $A$ and $B$. First, it is clearly a class that is definable from $F$ and $G$. Second, it is a function from $A$ to $B$. Note that if $a\in A_n$, then $G(F(a))\in A_{n+1}\subset A^+$, and so $F(a)$ is not $G^{-1}(a')$ for any $a'\in A-A^+$. Thus, the function $H$ is injective. Secondly, if $b\in B$ and $G(b)\in A_n$, then it must be that $n\geq 1$ and so $b=F(a')$ for some $a'\in A_{n-1}\subset A^+$, putting $b\in\ran(H)$. Otherwise, $G(b)\notin A^+$, and so again $b\in\ran(H)$. So $H$ is a bijection. QED
Some of the other proofs can also be formalized for classes, if one simply uses the sequences as I did here for iterating.