There are two ways double kick parts can be played on on a kit. The first would be to have two bass drums as part of your set up, each with it's own kick pedal. In this set up you would have a bass drum in the postion you would normally expect along with an additional bass drum to the left of this. The snare drum generally goes between the two pedals and toms and cymbals set up as normal around this. There are a couple of options for placing the hi hat. A standard hi hat pedal will work but will be further off to the left of the kit than usual due to the additional bass drum. There are Remote Hi Hat Stands where the top half of the stand and the pedal are separated and a cable is used to connect them, allowing the hi hat to be positioned a little easier. Another common option to bring the hi hat in closer is to fold up the legs of the hi hat stand and clamp it to the additional bass drum.
The second way to play double kick parts would be to have one bass drum with a Double Kick Pedal. These are ideal when you prefer the single kick set up, are limited for space or are gigging regularly using other peoples kits and can't guarantee a two bass drum set up. With these there is a pedal attached to your bass drum as normal that has two beaters and is connected to a 'slave pedal' by a cross which sits to the right of your hi hat stand. These can also present a slight issue of pushing the hi hats further away but there are many options to get around this. As well as those discussed above you can also get Two Legged Hi Hat Stands that accomodate these pedals really nicely.
Each of the above methods has its pro's and con's. Using two bass drums gives a more natural sound as with a single bass drum the notes will be cut short when the slave pedal is used. The double pedal is far more compact, easier to set up and requires less alterations to your kit to use. I personally use a double kick pedal but that is because I don't play a huge amount of double kick and was quite happy with my set up when I decided to add one and didn't want to make huge changes.
In terms of notating double kick both methods are pretty much identical but require a slight alteration to the wording of our Notation Guide. In our sheet music we notate for a Standard Bass Drum and a Second Bass Drum. It would make more sense to consider the Standard Bass Drum to be your right foot and the Second Bass Drum to be your left foot. The reason I have used that wording is because sometimes an alternate bass drum sound is used so it gives more options for notation.
In most cases of double kick notation the ordering of the feet is important. They follow the same rules as the hands where strong beats such as numbers and '+'s should be played with the right and weaker beats such as 'e's and 'a's should be played with the left.
