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Consider a reductive group $G$ over an algebraically closed field $K$ of characteristic $0$. I would like to consider the space $X$ of all $G$-conjugacy classes in $G$. Does the space $X$ have some nice geometric structure? (For instance, is it a variety? And/or if I take $K = mathbb{C}$ is the space $X(mathbb{C})$ Hausdorff? (It seems not to me, but I would like to ask)

Beware: I have not included the condition that I look at semi-simple conjugacy classes.

What you look it at is exactly the quotient of the representation variety $$Hom({mathbf Z},G)$$ of representations from the integers ${mathbf Z}$ to $G$, by the $G$-action via conjugation.

The quotient of the representation variety is in general not a variety, however you can look at the GIT-quotient, which is the character variety
$$X({mathbf Z},G)=Hom({mathbf Z},G)//G.$$
It is the Hausdorffification of the actual quotient. In the case $G=GL(n,{mathbf C})$ it is constructed by identifying two representations when all their traces agree. In your case this just means that you identify matrices if the traces of all their powers agree.

As already noted, the quotient need not be even $T_1$. For example, $SL_2(mathbb{C})/SL_2(mathbb{C})$ is homeomorphic to $mathbb{C}$ with double points at $pm 2$. This quotient is not $T_1$ since two of those points are not closed.

Here is a proof:

For any $tin mathbb{C}$, define $epsilon_t:=left(begin{array}{cc}t&-1\1&0end{array}right)$. Take any $Ain SL_2(mathbb{C})$ so that its trace is not equal to $pm 2.$ Then it has two distinct eigenvalues determined by its trace (use the quadratic formula on the characteristic polynomial to see this). Consequently, there is a matrix $g_A$ such that $g_A A g_A^{-1}=epsilon_t$. Consequently, each of these conjugation orbits is closed. If its trace is $pm 2$, then $A$ is either conjugate to $epsilon_t$ or equals $pmmathbf{1}$. In the former case, $epsilon_t$ is conjugate to one of $left(begin{array}{cc}pm 1& 1\ 0& pm 1end{array}right)$. Conjugating by $left(begin{array}{cc}n&0\0&frac{1}{n}end{array}right)$ then gives $left(begin{array}{cc}pm 1& frac{1}{n^2}\ 0& pm 1end{array}right)$. Letting $nto infty$ we see that $pm mathbf{1}$ is in the closure of the orbit of $epsilon_t$. $Box$

In general, $G/G$ will not be ``nice'' for complex reductive groups $G$ (unless $G$ is abelian).

However, $G/G$, in this generality, is homotopic to the GIT quotient $G/!!/Gcong T/W$, where $T$ is a maximal torus in $G$ and $W$ is the Weyl group. And in turn $G/!!/G$ is homotopic to the corresponding quotient $K/K$ for a maximal compact $K$ in $G$.

And, if $K$ is simply connected, its Weyl alcove is homeomorphic to $K/K$. Therefore, $K/K$ is homeomophic to a closed ball in this case.

Examples:

$SU(2)/SU(2)cong [-2,2]$

$SU(3)/SU(3)cong $