First thing first:
There is no way to cheaply do this in Java only. There are some chips that can execute Java, but I haven't heard of any which are accessible for beginners.
So it's either C or the Arduino's language if you go for that route.
Next, you're going to need a power source for the bell. This will depend on the bell and will make a major impact on the circuit you'll be making. You'll probably need a relay and protection circuit for it which will control the bell which will be powered from a separate power source (like for example mains power). You'll need to look up specifics for Arduino's (or whatever you want to use) pins and get a relay which can be triggered by such power source.
Next, you'll need Ethernet or USB connection. Matt already recommended USB to RS-232 chips and that is an easy way to solve the problem. The chip will appear as a serial port and all you'll need to do is make a Java program which will send data to the port to the microcontroller.
The other more complicated option would be to use Ethernet. For Arduino, there's an already existing Ethernet shield which is supported by its IDE. You could also pick up another Ethernet board like this one or this one. The positive side of Arduino shield is that you get to use existing library with it that could prove to be helpful. For a third-party solution, you'll have to write the code yourself.
Also a bit of comparison between Arduino, bare AVR chips or PIC.
First, today's PIC and AVR chips are more or less similar in price and capabilities. Some people will recommend PIC and some will recommend AVR. There are countless raging fanboys on each side, but my advice is to ignore them. If you're planning to seriously get into the microcontroller world, it would be good idea to get familiar with both families, because they are both very popular. The chips are generally programmed in C or assembly and there are many development tools (some of them free) available for both lines.
Next we have Arduino. First, it's not a chip. It's a development board line and that is an important difference. It uses AVR chips inside and a special bootloader to make them easier to program. With that you have packaged a nice IDE which uses a C++ like language for programming and may make some things easier for a beginner. Recently it has gained enormous popularity with beginners. Its main problem is that it hides some operational concepts of the chi[ from the user and at one point you may get the feeling that you're fighting the IDE and the language.
On the other hand, we have also the PICAXE which are basically PIC version of Arduino. They also have IDE and programming language (a BASIC derivative) and bootloader. As far as I can see PICAXE boards aren't as popular as Arduino for some reason, but I don't have any experience with them, so I can't elaborate why.
So to sum this up: If you want a single project, consider Arduino or PICAXE. This way, you won't have to pay for setup costs (like programmer, time needed for training and board design and so on) which working with single microcontroller brings.
If you plan to seriously work with microcontrollers in the future, consider doing this with an AVR of PIC chip. It'll be much more educational experience but it could get too complicated if you just doing a one-off project.