I have been reading other interesting posts written by fans of various TV shows based on comics, and I thought I would try my hand at it. So, here's what I think could happen next.
1. We Will Start Seeing More of Penguin Again.
This would be great, since so far Penguin and Nygma are my personal two favorite charters. Why is this the case? Simple, Jerome has had his last laugh. Personally, I thought that character would last a little longer before he bit the dust, but I saw this coming since they were not going to mess with The Joker's origin story that much.
Since we have not seen the Penguin much the last few weeks, they will most likely have him coming back in to take some of the spotlight. There is another reason that I feel this way though...
2. The one line in 'The Last Laugh.'
Penguin said that, "Someone like that has no interest in building this. He's not part of a tradition. Chaos for chaos's sake; perhaps I could use a laugh." With the inspiration of Jerome, there will be others over time that will be teased as having the qualities of The Joker. At some point it would make sense for Penguin to hire some of them for serious tasks, since let's face it, over time he has had some real whack jobs working for him.
3. We could see The Jokers
I am a product of the '90s, and remember the show Batman Beyond. In this there was a gang of Joker wannabes who used the same MO as he did.
With all the criminals on the street seen at the end, we could see a gang like this spring up in the city. The show has already explored this with The Red Hood Gang, and it would be interesting to see that again.
4. Barbra Could Seek Out Another
Let's face it, they were pretty chummy up until the end and she seemed a little torn up about him dying. If she has it in her head to get revenge for the death of her partner, no jokes please, she could seek out someone to help her take down Theo Galavan. Wouldn't that be a Season 2 ending?
These are several direction I could see plots within the show taking for its second season based on the events of 'The Last Laugh.'