We present high-resolution (0 ''.17. x 0.'' 14) Atacama Large Millimeter/submillimeter Array (ALMA) observations of the CO (6-5) line and 435 mu m dust continuum emission within a similar to 9 '' x 9 '' area centered on the nucleus of the galaxy NGC 5135. NGC 5135 is a well-studied luminous infrared galaxy that also harbors a Compton-thick active galactic nucleus (AGN). At the achieved resolution of 48 x 40 pc, the CO (6-5) and dust emissions are resolved into gas "clumps" along the symmetrical dust lanes associated with the inner stellar bar. The clumps have radii in the range of similar to 45-180 pc and CO (6-5) line widths of similar to 60-88 km s(-1). The CO (6-5) to dust continuum flux ratios vary among the clumps and show an increasing trend with the [Fe II]/Br gamma ratios, which we interpret as evidence for supernova-driven shocked gas providing a significant contribution to the CO (6-5) emission. The central AGN is undetected in continuum, nor is it detected in CO (6-5) if its line velocity width is no less than similar to 40 km s(-1). We estimate that the AGN contributes at most 1% of the integrated CO (6-5) flux of 512 +/- 24 Jy km s(-1) within the ALMA field of view, which in turn accounts for similar to 32% of the CO (6-5) flux of the whole galaxy.