Hi Daniel,
Oh boy. :D
The documentation you are looking for is here:
http://code.google.com/p/kryo/And here:
http://kryo.googlecode.com/svn/api/index.html
The SimpleSerializer methods write() an object to bytes and read() bytes to create an object. When you register a serializer for your Board class with Kryo, the object being written/read should be an instance of Board, not an instance of Class. IE, use new SimpleSerializer<Board> and not new SimpleSerializer<Class>.
In your write() method you need to write the data for each field that makes up a Board instance. In the read() method you need to create a new instance of your class, read the data for each field, and assign it to your instance. This is the manual way of serializing an object using Kryo.
Kryo has serializers that have various generic mechanisms to write and read data for objects without having to write code manually for each field. For networking, the serializer that is usually used is FieldSerializer:
http://kryo.googlecode.com/svn/api/index.html?com/esotericsoftware/kryo/serialize/FieldSerializer.html
If your Board class extends JPanel, it won't be easy to use FieldSerializer because JPanel has a bunch of fields that FieldSerializer will find and want to serialize. You can tell FieldSerializer to ignore specific fields or ignore all private fields. However, you would be better off decoupling your UI from your data, and then you could more easily send your data.
You could go even further and only send updates to the game board, rather than sending the entire game board every time anything changes. However, it is generally best to get something working first, then later you can work on improving the design.
-Nate