How would I create a desktop application that can see my online banking information?

I agree with Nelek in that it's probably a very bad idea - if only because you need to store your access codes in your app somehow, and that is effectively handing control of your bank account to anyone who can run your app! Still wanna risk it if you leave your lappie open at your best mates house?

Secondly, it's unlikely to work for long: banks change their login pages often both so that can advertise to you more, and to make it harder for script kiddies to hack accounts. They don't provide APIs for security reasons.

But, if you must, it's possible: a WebBrowser Control will let you control browsing to the site,. and you can supply the login info as if it was you typing. Just remember that if you make a mistake in your code, that's real money you are dealing with - and your real money as well! The bank isn't going to reimburse you for your cockup, even if you do accidentally send all your money to a Cats Home ...
And also remember that if you get a virus, that's just offering your money up to naughty people on a plate if you do this!

I'd have to agree with Griff and Nelek. This is a bad idea:

1) If you make mistakes - it could cause real problems.
2) If people get hold of your code/data that could cause issues.
3) I very much doubt the bank make their API public, your fiddling around trying to interact with it could conceivably be seen as a hacking attempt and get you in trouble.

I agree with all the above advice.

However, most banks provide an interface that will allow you to download transactions, so you can capture that data and do the budgeting offline. Contact your bank to see what facilities it offers.