You can expand confortably in a parameter when this parameter is small. Even if the coupling constant depends on the energy scale, one can expand in terms of the coupling constant at a fixed energy scale. This is a good thing to do only if this number is small. In $\phi^4$-theory (in four dimensions), the coupling is small in the IR but big in the UV: perturbation theory will tell you nothing interesting in the UV (and it is quite likely that $\phi^4$ theory does not exist in the UV). In QCD, the coupling constant is small in the UV and perturbation theory in this regime can be done and is done and is extremely ueful in this regime. It just becomes inadequate to understand the IR physics. Large $N$ limit is another perturbative scheme which is valid in a different range of parameters of the theory: it is useful for some questions, not for others.