Basically an electric ukulele is a ukulele with steel strings, which is amplified using an electro-magnetic pickup similar to those used in an electric guitar and other electric instruments. So far as I know the only such instruments that are commercially produced are the Konablaster in the USA, and the Risa electric ukuleles in Germany. However the Risa ukuleles should be banned because they use a Low G tuning, and a ukulele has that high G string which really annoys guitarists and just about anyone else who plays a musical instrument. And that's the way it should be!
So that leaves only the Konablaster as a possiblity. The other way of obtaining an electric ukulele is to make it, or have it made by a luthier, personally I made mine with some help from a luthier. The advantage of this is that you can choose what parts to fit, especially when it comes to pickups: the Konablaster comes with a single coil pickup and the Risa, apart from not being a true ukulele, is equipped with two lipstick pickups, which is not necessarily ideal for melodic death metal. I fitted a Mighty Mite Motherbucker double coil pickup (Humbucker), which I find because of its high gain gives the clean sound a kind of round ukulele-ish sound, rather than a sharp guitar-ish sound, and of course for distortion it is excellent.
And why on earth would anyone want to play electric ukulele?
The main thing about the ukulele is that it has "uke attitude", especially when electric. It is just really cool to be able to say in a superior way to people: "that's not a mandolin/banjo/toy guitar, that's my electric ukulele". That is the one thing to remember about the ukulele: IT IS NOT A GUITAR! And yet you can do just as many amazing things on it as on a guitar, and it's more impressive because people are not used to "ukulele heroes/shredders", whereas "guitar heroes" are a relatively common breed.
So that is the main point of being an electric ukulele player: being weird, a bit like bass players, except that ukulele players have a more chords and solo role than the bass, and therefore they are more noticed (although I have to say that this is pure theory because I don't know of any bands apart from the one I play in who have ukulele players who play solos).
I'm not really explaining why you should play electric ukulele rather than acoustic ... Let's say that the rarity of electric ukuleles even in the ukulele world enhances the weirdness factor. But it also allows one to use effects such as distortion or overdrive without the annoyance of feedback, and use techniques such as tapping. This is especially useful for playing Heavy Metal and Hard Rock type styles.
And how is the electric ukulele supposed to be played?
Once again I will remind you: The ukulele is NOT A GUITAR. This means that all form of plectrum/pick should be kept at a safety distance of 100m from ukuleles, to help any uncivilised people resist the temptation to even attempt to touch the ukulele strings with a pick: it sounds horrible and goes against the principles of "uke attitude". The only way to play true ukulele is with the fingers, and remember, no Low G Tunings! The Low G Tuning was invented by guitarists who are too far gone in the mainstream guitar attitude to accept the possibility that any stringed instrument could dare be tuned differently from a guitar.