There are loads of different development roles in banking. The highest paying ones tend to require some sort of quant experience. That's degree+ level maths with business AND development skills. It's a big ask.
I'll assume you don't have strong financial maths, otherwise you'd probably already know the answer to the question :)
Financial companies vary, there are a lot of them and they have different criteria. Actually the same criteria but in different quantities. These are:
1) Business area experience. Be it general investment banking, fixed income, commodities, front office, risk, what ever, some roles need more business experience than others. Generally the closer you work with the business the more business experience will be required. This can be a classic catch 22, but there are ways around it.
2) Technical skills. If you can demonstrate good technical skills this can be enough. Certain companies actually prefer people with less business experience because they can pay them less, but it can act as a good first step to better things. This tends to be software companies and consultancies. With in the banks themselves, it really depends on the bank, and with in the bank, the actual team you are interviewing for. Really you could interview for two teams in the same bank on the same day and it would feel like two totally different companies.
3) Interpersonal skills. This can swing it, I've strong technical people with good business knowledge fail in an interview because they are not assertive enough, or too assertive, or smell... Handy tip, the people who interview you are people. This means they're subject to the whole gamut of emotional stuff everyone else is. This means they can take an irrational dislike to you (or a liking to you) and there's just nothing you can do about it besides not letting it get you down.
4) Contract type. Finally banks hire either contract or perm staff. Perm staff generally have to jump through more hoops to land the role than contract staff, but if you get a contract, you're then gaining experience and you may get the opportunity to switch to perm if that's what you want. So I'd consider applying for contract positions to start with.
As for where to find them. I use job web sites, spam 20 or so roles, this puts me in touch with a load of agents and then they do the rest. Some banks have a public portal for job applications, so it's worth vising every bank web site and looking for a jobs section.
Finally look at consultancies. I mentioned this above. I don't want to list a load of names here, but there are financial consultancies, financial software houses, then things like brokers and out source companies. Outside of those there are hedge funds which are like mini investment banks. Each of those will have different criteria too, and it will give you another larger set of companies to approach outside of the dozen or so banks.