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Yes, you can do this using polar decomposition. We can also consider $\phi$ to be a normal linear functional on $A^{**}$, and there is a positive $\omega \in A^*$ and a partial isometry $v \in A^{**}$ such that $\phi(x) = \omega(vx)$ for all $x \in A$. (I'm sure this is in volume 1 of Takesaki, probably also in Pedersen.) We have $\|\omega\| = \|\phi\| \leq 1$, so we can apply GNS to $\omega$as you say the vector $\xi$ achieves the conclusion for $\omega$