We present Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS) and Wide Field Camera 3 (WFC3) observations of SDSS J1029+2623, a three-image quasar lens system produced by a foreground cluster at z = 0.584. Our strong lensing analysis reveals six additional multiply imaged galaxies in addition to the multiply imaged quasar. We confirm the complex nature of the mass distribution of the lensing cluster, with a bimodal dark matter distribution which deviates from the Chandra X-ray surface brightness distribution. The Einstein radius of the lensing cluster is estimated to be theta(E) = 15.2 +/- 0.5 arcsec for the quasar redshift of z = 2.197. We derive a radial mass distribution from the combination of strong lensing, HST/ACS weak lensing and Subaru/Suprime-cam weak lensing analysis results, finding a best-fitting virial mass of M-vir = 1.55(-0.35)(+0.40) x 10(14) h(-1) M-circle dot and a concentration parameter of c(vir) = 25.7(-7.5)(+14.1). The lensing mass estimate at the outer radius is smaller than the X-ray mass estimate by a factor of similar to 2. We ascribe this large mass discrepancy to shock heating of the intracluster gas during a merger, which is also suggested by the complex mass and gas distributions and the high value of the concentration parameter. In the HST image, we also identify a probable galaxy, GX, in the vicinity of the faintest quasar image C. In strong lens models, the inclusion of GX explains the anomalous flux ratios between the quasar images. The morphology of the highly elongated quasar host galaxy is also well reproduced. The best-fitting model suggests large total magnifications of 30 for the quasar and 35 for the quasar host galaxy, and has an AB time delay consistent with the measured value.