There are two issues mixed up in your question.
First, there is a preferred cosmic rest frame: the frame of the cosmic microwave background. This does not imply any contradiction with the principle of relativity, because the rest frame is not an intrinsic property of general relativity, but just a property of the configuration of matter in the particular solution of general relativity we call home. There is a more detailed explanation of this point here: Is the CMB rest frame special? Where does it come from?
The cosmic rest frame is used to set a reference for many questions for cosmology. For example, to define the age of the Universe, we mean what is the time between now and the Big Bang as measured in the cosmic rest frame. When we talk about the expansion rate, we mean how fast distances between galaxies -- as measured in the cosmic rest frame -- are receding from one another per unit time -- as measured in the cosmic rest frame.
Second, you are correct that, strictly speaking, we should only talk about relative velocities in a locally intertial frame. In the context of cosmology, this means in practice that we should only talk about velocities of objects closer than about 100 Mpc away from us. For objects closer than 100 Mpc, the redshift is much smaller than one, and the effects of the expansion of the Universe can be ignored to a good approximation (or in other words, we can ignore the curvature of spacetime on these scales).
For galaxies closer than 100 Mpc, the component of the velocity due to the expansion of the Universe from us is determined by Hubble's law, $v = H_0 D = c z$; since $z \ll 1$, then $v \ll c$. Above about 100 Mpc, the redshift starts to become comparable to one, and the velocity you would infer from Hubble's law becomes relativistic. These very distant galaxies are the ones that are not in our same locally inertial frame, and you are correct that we shouldn't really talk about a velocity relative to them at all. To summarize: for galaxies close enough that it makes sense to talk about a relative velocity, the component of the relative velocity due to the expansion of the Universe is always much less than $c$. (This doesn't rule out that some object might be nearby and whizzing through space at a relativistic speed).
When people talk about the velocity of galaxies receding away from us at or above the speed of light due to the expansion of the Universe, more often than not what they are doing is extrapolating Hubble's law $v=cz$ outside its regime of validity $z\ll 1$. So long as you know what you mean, this is ok-ish, but it can be extremely confusing when you are first learning the subject.
