What is the best way of breaking a mathematical development into a sequence of articles?
Each journal article should "tell a story". Break the thesis up into coherent stories.
It looks like you're already doing that. The first article concentrates on generalizing the classical theorem, and the second gives the results that are relevant to current research.
I would suggest just putting the groundwork you need for the generalization of the classical theorem into the first article (unless you can save a lot of space by developing some of the groundwork for the second article at the same time as the groundwork for the first). And you should definitely mention the second article to provide more motivation for the groundwork. But putting groundwork that's not needed in this first article will detract from the "story".
In the second article, you should include a quick review of the groundwork developed in the first article, and refer the reader to the first article if they want more details. Then add the extra groundwork needed for the second article, and proceed to the results of the second article.
One comment from experience: if you put a result that does not fit the "story" into a paper, when people read your paper, they are very likely not to even notice this result. So it's much better to put it in a separate paper.
On the other hand, as you can tell from the comments here, many people consider it bad form to spread one story over two papers. So you should split a long result into several papers only if you can formulate it as several related "stories".
You have several considerations here. First, where will you publish? It sounds lengthily, and a two-parter will take up valuable real estate, so I'll start by suggesting you send some pre-prints/feelers out beforehand to journals to gauge interest. These pre-submission inquiries can save you tons of time and may even grease the wheels. It would help if both parts were published together in the same journal, but I've seen cases where parts I and II appear in different publications, sometimes with considerable lapse in time. Working the field beforehand will give you an idea how this is going to go.
In terms of a standard format, I've never heard of one, but in general I find some consistency. One is that they are often named 'Theory of things I: A solution to bob's paradox'. Then 'Theory of things II: Extension to Wendy'. Besides the conjecture (I've heard) that titles with colons get higher citation counts, this course adds a certain gravitas that can't be overlooked.
Next, both articles need to be able to stand alone as a piece of work. It's not feasible to do part I as background and II as the theory or solution, but it sounds like you're already preparing for this. Lay out your background and first contribution in part I. Hopefully, part I will be very intriguing and whet interest for part II. Then, in part II, refer to pt I extensively but don't be shy about reprinting very important parts that are integral to comprehending the current (pt2) material. For example, an equation might be reprinted (and referenced) while a general idea might just be referenced to your original paper. If you've played the pre-submission inquiry well, you might already have some leniency on space constraints, so use them.