We present an analysis of the molecular and atomic gas emission in the rest-frame far-infrared and sub-millimetre, from the lensed z=2.3 sub-millimetre galaxy SMM J2135-0102. We obtain very high signal-to-noise detections of 11 transitions from 3 species and limits on a further 20 transitions from 9 species. We use the 12CO, [CI] and HCN line strengths to investigate the gas mass, kinematic structure and interstellar medium (ISM) chemistry, and find strong evidence for a two-phase medium comprising a hot, dense, luminous component and an underlying extended cool, low-excitation massive component. Employing photo-dissociation region models we show that on average the molecular gas is exposed to a UV radiation field that is ~1000 x more intense than the Milky Way, with star-forming regions having a characteristic density of n~10^4 /cm^3. These conditions are similar to those found in local ULIRGs and in the central regions of typical starburst galaxies, even though the star formation rate is far higher in this system. The 12CO spectral line energy distribution and line profiles give strong evidence that the system comprises multiple kinematic components with different conditions, including temperature, and line ratios suggestive of high cosmic ray flux within clouds. We show that, when integrated over the galaxy, the gas and star-formation surface densities appear to follow the Kennicutt-Schmidt relation, although when compared to high-resolution sub-mm imaging, our data suggest that this relation breaks down on scales of <100pc. By virtue of the lens amplification, these observations uncover a wealth of information on the star formation and ISM at z~2.3 at a level of detail that has only recently become possible at z<0.1, and show the potential physical properties that will be studied in unlensed galaxies when ALMA is in full operation. (Abridged).
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