If I understand well, you would like to reverse a Java applet in order to plug your own tool on a specific API (which is incidentally a trading platform).
Anyway, what you need to use is a Java Decompiler in order to recover as much possible from the
*.class files you got from the application you want to understand. There are several Java Decompilers that you can find, here is a small list (from the most featured to the least):
You can find more information by looking to this similar question on SO.
If its web based, you may not have to reverse the Java app.
Have you tried tools like Fiddler - http://www.telerik.com/fiddler - to capture web traffic that occurs when you execute a trade?
You could figure out what parts of the API the browser is actually calling when certain things are executed by capturing that in Fiddler.
You could then write a program in python to act as you in the browser.