I think a Tesla coil may be the wrong approach. For EMP, you need a brief high energy pulse, whereas Tesla coils provide extremely high voltage output, but with low power.
I don't remember seeing any DIY EMP devices on the web, but I have seen things like rail guns, magnetic can crushers and magnetic coin shrinkers. If you can find some of those, I wouldn't be surprised if there's a link on one of those websites to an EMP project. They all share the same key ingredient - a capacitor bank that can store a huge amount of energy and release it extremely quickly.
Plan B is just to google DIY EMP and see what comes up.
P.S. Now that I think about it, I seem to remember seeing some ideas for smallish explosives-based EMP weapons, but I'd avoid those. Aside from legal issues, a high energy capacitor bank based solution is likely to be (at least marginally) safer. Unlike a stick of dynamite, a capacitor bank isn't supposed to explode. If it does, the result is likely to be similar though, so... serious safety precautions are needed whatever approach you go for.