This is the fifth in a series of papers studying the astrophysics and cosmology of massive, dynamically relaxed galaxy clusters. Our sample comprises 40 clusters identified as being dynamically relaxed and hot in Papers I and II of this series. Here we use constraints on cluster mass profiles from X-ray data to test some of the basic predictions of cosmological structure formation in the Cold Dark Matter (CDM) paradigm. We present constraints on the concentration-mass relation for massive clusters, finding a power-law mass dependence with a slope of κ m = −0.16 ± 0.07, in agreement with CDM predictions. For this relaxed sample, the relation is consistent with a constant as a function of redshift (power-law slope with 1 + z of κ ζ = −0.17 ± 0.26), with an intrinsic scatter of σ ln c = 0.16 ± 0.03. We investigate the shape of cluster mass profiles over the radial range probed by the data (typically ∼ 50 kpc-1 Mpc), and test for departures from the simple Navarro, Frenk & White (NFW) form, for which the logarithmic slope of the density profile tends to −1 at small radii. Specifically, we consider as alternatives the generalized NFW (GNFW) and Einasto parametrizations. For the GNFW model, we find an average value of (minus) the logarithmic inner slope of β = 1.02 ± 0.08, with an intrinsic scatter of σ β = 0.22 ± 0.07, while in the Einasto case we constrain the average shape parameter to be α = 0.29 ± 0.04 with an intrinsic scatter of σ α = 0.12 ± 0.04. Our results are thus consistent with the simple NFW model on average, but we clearly detect the presence of intrinsic, cluster-to-cluster scatter about the average.