How you introduce code katas into your workplace depends very much on what you want to get out of them.
In my case I wanted to encourage the use of Test Driven Development so I ran a Cyber-Dojo. With this sort of exercise, the emphasis is not on the code itself, but on the process of writing the code.
We spent an afternoon, in pairs, repeating the same kata, but under different conditions. We started with all groups doing one exercise at the same time. This provided a baseline.
We then discussed some of the basic principles of TDD, had everyone change partners and repeat the same kata. We repeated the same kata to de-emphasise the generation of code and instead concentrate people on the process of naming test cases and the Red/Green cycle.
Then we repeated the kata again, but roughly every 10 minutes one person in each group would move to another group, simulating the rather fluid team environments we often find ourselves in these days.
In the final iteration, we had both partners change every 10 minutes or so into different groups. This helped to demonstrate that with TDD, even the handover from one team to a completely different one needn't necessarily be too painful, since the project should only every be one Red/Green cycle from working.
The interesting thing was, there were few people who had done any TDD before the session, but what TDD knowledge there was rapidly spread until by the final iteration through the kata, most people were thinking in a TDD way or could at least appreciate why it might be beneficial.
People generally said that the afternoon was both fun and informative and we are now looking at other ways to use Cyber-Dojo at my workplace.
Cyber-Dojo, written by Jon Jagger is works incredibly well for this sort of exercise. It is a web based integrated environment for doing deliberate practice of TDD and learning about team dynamics. It has lots of kata's selected specifically to help people concentrate on the process of TDD and not the problem. It also supports a range of languages, from Python and Ruby to Java and C++.
The best thing is, after doing a kata you can then go back and look at the red/green progression (or maybe not *8') of each of the groups participating. It's traffic lights are a great way to visualise how the TDD process works.
If you want your own CyberDojo server, the whole project can be found at github and there is even a Turnkey Linux appliance virtual machine linked from there, which means that assuming you already have VMware player or VirtualBox installed, you can be up and running within a few minutes of downloading the appliance!