Context: Star formation at earlier cosmological times took place in an interstellar medium with low metallicity. The Large Magellanic Cloud (LMC) is ideally suited to study star formation in such an environment. Aims: The physical and chemical state of the ISM in a star forming environment can be constrained by observations of submm and FIR spectral lines of the main carbon carrying species, CO, C i and C ii, which originate in the surface layers of molecular clouds illuminated by the UV radiation of the newly formed, young stars. Methods: We present high-angular resolution sub-millimeter observations in the N159W region in the LMC obtained with the NANTEN2 telescope of the 12CO J = 4 to 3, J = 7 to 6, and 13CO J = 4 to 3 rotational and [C I] ^3P1-^3P0 and ^3P2-^3P1 fine-structure transitions. The 13CO J = 4 to 3 and [C i] ^3P2-^3P1 transitions are detected for the first time in the LMC. We derive the physical and chemical properties of the low-metallicity molecular gas using an escape probability code and a self-consistent solution of the chemistry and thermal balance of the gas in the framework of a clumpy cloud PDR model. Results: The separate excitation analysis of the submm CO lines and the carbon fine structure lines shows that the emitting gas in the N159W region has temperatures of about 80 K and densities of about 104 cm-3. The estimated C to CO abundance ratio close to unity is substantially higher than in dense massive star-forming regions in the Milky Way. The analysis of all observed lines together, including the [C ii] line intensity reported in the literature, in the context of a clumpy cloud PDR model constrains the UV intensity to about χ ≈ 220 and an average density of the clump ensemble of about 105 cm-3, thus confirming the presence of high density material in the LMC N159W region.