Based on the comments I'm going to assume that the measure you're interested in is the pushforward of Haar measure to the conjugacy classes of $\text{SU}(3)$ (I know nothing about why you should expect this). There are a lot of things you can say about this measure without writing down an explicit formula for it. In particular, you can compute the moments of various random variables with respect to this measure.
The basic observation is that the map sending a (complex, unitary, finite-dimensional) representation of $\text{SU}(3)$ to its character induces a homomorphism $\chi$ from the representation ring of $\text{SU}(3)$ to the algebra of class functions on $\text{SU}(3)$, and by Peter-Weyl the span of the image is dense. Moreover, we know the integral over $\text{SU}(3)$ of the character of every representation: it's the dimension of the invariant subspace. So to compute moments of class functions on $\text{SU}(3)$ (random variables on the conjugacy classes) it suffices in principle to understand invariant subspaces of tensor products of representations of $\text{SU}(3)$ (given that you know how to express your class functions in terms of characters).
For example, suppose you're interested in the moments of twice the real part of the trace of a random element of $\text{SU}(3)$. This is the character of the representation $W = V \oplus V^{\ast}$ where $V$ is the defining representation. The $n^{th}$ moment is then precisely the dimension of the invariant subspace of $W^{\otimes n}$, and to compute this it suffices to count the number of walks of length $n$ in a Weyl chamber from the origin to itself by highest weight theory. The first few moments are $$1, 0, 2, 2, 12, 30, 130, ....$$
This is A151366 in the OEIS. There is a generating function given there but it involves hypergeometric functions, so like Marty I don't expect a particularly nice explicit formula for the corresponding measure.
(Doing the above for $\text{SU}(2)$ you reduce the problem to computing the moments of a single random variable, the trace. The corresponding representation is just the defining representation, the Weyl chamber is $\mathbb{Z}_{\ge 0}$, and walks of length $n$ in the Weyl chamber from the origin to itself are counted by Catalan numbers, which are the moments of the Sato-Tate measure.)
Edit: Emerton says in the comments that the trace of an element of $\text{SU}(3)$ takes values in $[-3, 3]$ and doesn't determine the conjugacy class. In fact, it takes values in the ball of radius $3$ in $\mathbb{C}$ and it does determine the conjugacy class!
Let $g \in \text{SU}(3)$ have trace $z$. Recall that its eigenvalues $\lambda_1, \lambda_2, \lambda_3$ are unit complex numbers satisfying $\lambda_1 \lambda_2 \lambda_3 = 1$. It follows that $$\lambda_1 \lambda_2 + \lambda_2 \lambda_3 + \lambda_3 \lambda_1 = \overline{\lambda_1 + \lambda_2 + \lambda_3}$$
hence that $\text{tr}(g)$ determines the characteristic polynomial of $g$. So to understand the distribution of the eigenvalues of a random element of $\text{SU}(3)$ it suffices to understand the moments of the real and complex parts of the trace. The real part is handled above; the complex part comes from the virtual representation $V \ominus V^{\ast}$ and so one must count walks in a Weyl chamber with certain weights $\pm 1$.