It is very important to improve the thermal conductivity of polymer nanocomposites (PNCs) to widen their application. In this work, by employing reverse nonequilibrium molecular dynamics simulations in a full atomistic resolution, we systematically investigated the effect of the chemical grafting of carbon nanotube (CNT) on the thermal conductivity of PNCs. First, the interfacial thermal conductivity is proportional to the grafting density, while it first increases and then saturates with the grafting length. Meanwhile, the intrinsic in-plane thermal conductivity of CNT drops sharply as the grafting density increases. Combined with effective medium approximation, the maximum overall thermal conductivity of PNCs appears at an intermediate grafting density because of these two competing effects. In addition, two empirical formulas are suggested, which quantitatively account for the effects of grafting length and density on the interfacial and parallel thermal conductivity. Secondly, the thermal resistance between the CNTs gradually decreases with the increase of the grafting density and grafting length, which can be well described by an empirical equation. The heat transfer process from one CNT to another can be well described by a thermal circuit model between the CNTs. Thirdly, a stronger enhancement of the thermal conductivity is realized when chains are grafted at the end atoms of CNTs. Under deformation, the orientation of both the chains and the CNTs improves the thermal conductivity parallel to the tensile direction, but reduces the thermal conductivity perpendicular to it. We have quantified the contribution of the polymer alignment and the CNT alignment to the anisotropy of thermal conductivity. In general, computer simulation is shown to have the capability to obtain some fundamental understanding of PNCs, in hopes of providing some design principles for fabricating high performance PNCs.