This is the first in a series of papers on the properties of ultra-diffuse galaxies (UDGs) in clusters of galaxies. We present an updated catalog of UDGs in the Coma cluster using \textit{g}- and \textit{r}-band images obtained with Hyper Suprime-Cam (HSC) of the Subaru telescope. We develop a method to find UDGs even in the presence of contaminating objects, such as halos and background galaxies. This study expands upon our previous works that covered about half the area of the Coma cluster. The HSC observations covered the whole Coma cluster up to the virial radius and beyond (an area twice larger than the previous studies) and doubled the numbers of UDGs ($r_{\rm eff, r} \geq 1.5$ kpc) and sub-UDGs ($1.0 \leq r_{\rm eff, r} < 1.5$ kpc) to 774 and 729 respectively. The new UDGs show internal properties consistent with those of the previous studies (e.g., S\'ersic index of approximately 1), and are distributed across the cluster, with a concentration around the cluster center. The whole cluster coverage clearly revealed an excess of their distribution toward the east to south-west direction along the cluster center, where Coma connects to the large scale structure, and where a known substructure exists (the NGC4839 subgroup). The alignment of the UDG distribution along the large scale structure around Coma supports the interpretation that most of them lie at the distance of the Coma cluster and the NGC4839 subgroup.
