One might hope to handle proper classes as objects by working in one of the standard second-order set theories. For example, there is Gödel–Bernays set theory GBC, which has classes as objects, and includes familiar class axioms, including the first-order class comprehension axiom, which asserts that every first-order definable property defines a class. Kelley–Morse set theory KM extends this comprehension axiom to allow second-order definitions, where the second-order quantifiers range over classes. There is an emerging literature classifying various second-order theories over GBC, analyzing the strength of various principles such as the existence of forcing relations for class forcing, the existence of first-order truth predicates and transfinite class recursions. The principle CC or class comprehension is commonly considered as a strengthening of KM.
So, with these second-order theories, one can handle the existence of classes.
An immediate problem for your proposal, however, is that a measure on the classes is not itself a class object, but a third-order object, a meta class. Specifically, a measure would be a map associating every class with a real number. For this reason, even the second-order set theories do not seem able in general to handle arbitrary measures.
One way around this issue (as suggested by David Roberts in the comments), which will work when the collection of measurable classes is comparatively small, is to consider measures that are codeable entirely into a single class. For example, one could consider the underlying collection of measurable sets as the set of sections $M\subset V\times V$, where sections are $M_x=\{y\mid (x,y)\in M\}$, and then consider the measure as a class function $\mu:V\to\mathbb{R}$. When the collection of measurable classes is indexable by $V$, then this approach works fine in the second-order frameworks such as KM. Probably one will usually want to augment KM with stronger axioms such as the class DC schema, which is not provable in KM. This method of coding into classes by sections is common in many other context — for example, one can construct class binary relations $\Gamma$ on Ord that code "metaordinals" higher than Ord, and then aim to undertake class recursions of length $\Gamma$. This is the underlying idea in the principle ETR.
Another natural way to handle larger class measures is to find definable measures, avoiding the need to code up the entire collection of measurable classes into a single class. These measures would be definable associations of every class $X\subseteq\newcommand\Ord{\mathrm{Ord}}\Ord$ with its measure
$$X\mapsto \mu(X).$$
And indeed, there are some extremely natural instances of this, which appear already in the set-theoretic literature.
Consider for example the club measure, defined on $X\subseteq\Ord$ so that $\mu(X)=1$, if $X$ contains a closed unbounded class $C\subseteq X\subseteq\Ord$, and $\mu(X)=0$, if $X$ is disjoint from a closed unbounded class $X\cap C=\varnothing$. This is a second-order definable map, since we say $X$ has measure one if there is a club class inside $X$, and measure $0$ if there is a club class disjoint from $X$. The positive classes are exactly the stationary classes, those that have nontrivial intersection with every club class.
One can prove in KM+CC that this measure is countably additive and indeed $\kappa$-additive for every cardinal $\kappa$. The main point is that if one has a $\kappa$-sequence of measure one sets $X_\alpha$, containing clubs $C_\alpha\subseteq X_\alpha$, then by CC one can choose particular such clubs $C_\alpha$, and the intersection of any set number of closed unbounded classes of ordinals remains club. So there is a single club class $C$ contained in every $X_\alpha$, and so $\bigcap_\alpha X_\alpha$ has measure one.
Meanwhile, an intriguing situation occurs with this measure without the CC axiom. Namely, without CC one cannot in general pick the club classes $C_\alpha$, and in some joint work with myself, Vika Gitman and Asaf Karagila, we proved that it is consistent with mere KM that this measure is not countably additive.
This observation can be taken as evidence that Kelley-Morse set theory is less robust than might have been imagined, and for such kind of uses of proper classes, one will often want to augment it with CC or much more.
