We present deep observations at 450 and 850{mu}m in the Extended Groth Strip field taken with the SCUBA-2 camera mounted on the James Clerk Maxwell Telescope as part of the deep SCUBA-2 Cosmology Legacy Survey (S2CLS), achieving a central instrumental depth of {sigma}_450_=1.2mJy/beam and {sigma}_850_=0.2mJy/beam. We detect 57 sources at 450{mu}m and 90 at 850{mu}m with signal-to-noise ratio >3.5 over ~70arcmin^2^. From these detections, we derive the number counts at flux densities S_450_>4.0mJy and S_850_>0.9mJy, which represent the deepest number counts at these wavelengths derived using directly extracted sources from only blank-field observations with a single-dish telescope. Our measurements smoothly connect the gap between previous shallower blank-field single-dish observations and deep interferometric ALMA results. We estimate the contribution of our SCUBA-2 detected galaxies to the cosmic infrared background (CIB), as well as the contribution of 24{mu}m-selected galaxies through a stacking technique, which add a total of 0.26+/-0.03 and 0.07+/-0.01MJy/sr, at 450 and 850{mu}m, respectively. These surface brightnesses correspond to 60+/-20 and 50+/-20 per cent of the total CIB measurements, where the errors are dominated by those of the total CIB. Using the photometric redshifts of the 24{mu}m-selected sample and the redshift distributions of the submillimetre galaxies, we find that the redshift distribution of the recovered CIB is different at each wavelength, with a peak at z~1 for 450{mu}m and at z~2 for 850{mu}m, consistent with previous observations and theoretical models.