Homemade particle accelerator

It is not so hard, but it won't be able to generate enough high energetic particles.

The best example for a particle accelerator is a CRT (cathode ray tube), which you can find in every CRT monitor or TV. It can generate around $40\rm\,keV$ electrons. (LHC generates $3.5\rm\,TeV$ protons, thus it is around a hundred million times stronger).

Only a particle accelerator isn't enough, if you want to make experiments with it, you need some analytic/measurements devices also. In the current accelerators, they are nearly so complex and costly as the main accelerating device.

There is also a device capable to be built in home, it is the Farnsworth fusor:

Maybe it is not a particle accelerator in the classical sense, it creates enough strong field to be able to fuse deuterons (although it is doing this with terrible efficiency, around $10^{-8}$). You can see a Farnsworth fusor scematic below:

_{(source: fusor.net)}

There is a whole community of home fusors which can be found here.

Well, there are a lot of different types of particle accelerators.

Besides the already suggested cathode ray tube, another rather small accelerator would be a cyclotron. The first one was built by Lawrence and Livingston, "a device about 4.5 inches in diameter used a potential of 1,800 volts to accelerate hydrogen ions up to energies of 80,000 electron volts." I guess it would be possible to get a high voltage power source and hydrogen, but really building a working device would probably take a long time.