Plex can be installed on a Synology DiskStation through the Package Center. The list of supported Synology products is located in the summary for the Plex package (the DS1512 is supported.)
Once you've installed it, and started it up, you point your web browser to:
From there, you're able to configure your library sections, giving them each the location of your media on the DS1512.
Once that's done, it'll scan those locations and build the library inventory. That part can take hours, depending on the size of your library. I left mine overnight on a Friday night, and it was done when I got back to it the following afternoon (completed sometime during the night.)
However, I found the sorting to be rather lacking. For example, I used iDentify to fill in the media information for my library, and it all shows up perfectly in iTunes. But, when I bring up a TV episode from Season 1 of a show in Plex, it will have some episodes from a completely different TV series and season listed as episodes for Season 1 in that first TV show. It was very annoying, and I didn't have the patience, nor time, to sit down and figure out why. Also, some of the TV shows or episodes didn't appear in the Plex Media Library even though the scan was complete (also annoying.)