Since free C-algebras don't exist, we can't give a concrete description of what relations are allowed. Instead, we need to give axioms that determine what collections of n-tuples $(a_1,\dots,a_n)$ in $A$ are allowed, where $A$ varies over all C-algebras. It is important that the elements not "know about" the ambient C-algebra, so ``a is in a separable C-algebra'' is not allowed.
Some of what is allowed is conditions like $0 \leq x \leq 1$ and $$ 0\leq\left[\begin{array}{cc} \mathbf{1} & x\\ x^{*} & \mathbf{1} \end{array}\right]\leq 1 $$ where $\mathbf{1}$ is in the unitization of $A$. Another fun example is ``x is hermitian and has spectrum contained in the Cantor set''.
I could drone on forever here, but I already did so here: ``C*-Algebra Relations'' in Mathematica Scandinavica 107, 43--72, 2010.