From the beginning, Gotham has been telling origin stories for the various characters that would eventually be part of the Batman universe. It has really been the first live-action telling of the story during the period from when Bruce Wayne's parents were killed until he becomes Batman. The show would have benefitted from a 10-season run to give David Mazouz time to grow up to a point where he could believably be Batman. Even though he got taller throughout the series's run, at age 19 (as he was during this season), he was still not very muscular, and he still looked like a teenager, so it was hard to buy him as being intimidating in the suit.
The final season picks up months after the events that ended season 4. For most of the season, up through episode 11, the show did its take on the "No Man's Land" story from the comics. It was a very different telling of that story from the version shown in The Dark Knight Rises, the final film of the Nolan Trilogy of movies. You must suspend disbelief to buy the story as a whole, given that a few bridges blowing up would not really be enough to keep help from coming in, getting people out, etc. That said, it was very well done and did provide a way to introduce Bane, played by Shane West, into the story. That part of the season was mostly about tying up that last storyline, introducing some new elements (such as the birth of Barbara Gordon, who will become Batgirl), and resolving storylines of the various villains.
The final episode does a time jump. It will not be giving anything away to say we see Batman in the final episode. That much was given away before the season even started. It was a decent series finale, but it did feel a bit too rushed. It should have been a two-hour finale with more interactions between the characters. After watching it again, I know why the showrunners made the choices in the final episode they did, but it may leave some people unsatisfied. The big controversy surrounding the final episode was recasting Selina and replacing Cameron Bicondova with Lilli Simmons (probably best known for the series Banshee). Bincondova has said that it was her choice not to play the older Selina, and to me, given Selina's role in the finale, it was not a huge deal.
Overall, I think the show did a good job with the shortened season and ultimately took the story where the writers and producers intended it to go. That said, I do think that because there were 10 fewer episodes than the usual 22 episodes, the writers were trying to pack a ton of material into the limited number of episodes to give all the characters a show that at least highlighted, if not was centered, around them, that it did not tell the story as well as it could have been told.
For those who get the blu-rays, the 12 episodes are on two discs. They are (thankfully) the blu-ray sets that allow you to play all and pick up where you leave off if you stop midway through an episode. The extras include deleted scenes from some episodes, a portion of the NY comic-con panel with some of the actors and producers, a 15-minute behind-the-scenes featurette on the final season, and a 38-minute featurette on various DC comic villains (not only from the show, but the other DC shows like Arrow, Flash, and Krypton). So, it has a pretty good amount of bonus content, especially considering how light some TV series sets being released on physical media these days are on bonus features.
Overall, it was a good show that had a pretty good run. I cannot say everyone will like how the show ended, but given that there was no guarantee that the show would even get a final season, I think they did the best they could with the number of shows they had to work with.