The multiplicity of the irreducible character of $S_n$ indexed by the partition $\lambda$ of $n$ in the action of $S_n$ on itself by conjugation is the coefficient of the Schur function $s_\lambda$ in the Schur function expansion of $1/(1-p_1)(1-p_2)(1-p_3)\cdots$, where $p_i$ is a power sum symmetric function. (This is an exercise in Chapter 7 of Enumerative Combinatorics, vol. 2.) This makes it very easy to compute the decomposition of the conjugacy action using Stembridge's SF package for Maple. For instance, I computed on my laptop the entire decomposition for $n=20$ in a couple of seconds.
Note added 1/10/14: Originally I inadvertently wrote $1/(1-p_1-p_2-p_3-\cdots)$ instead of $1/(1-p_1)(1-p_2)(1-p_3)\cdots$. This has been corrected. The "incorrect" symmetric function $1/(1-p_1-p_2-p_3-\cdots)$ is also interesting. It is equal to $$ \frac{\sum_{n\geq 0}h_n}{1-\sum_{n\geq 1}(n-1)h_n}. $$ See for instance http://www.math.miami.edu/~wachs/papers/chrom.pdf and let $t\to 1$.