Flatwater cartwheeling is probably the most difficult kind of cartwheel to do because all the energy and rotational momentum, is provided solely by the paddler. You will need a kayak that is fairly slicey and fits you well.
A double pump initiates the move. The easiest way to do this is to paddle forwards at a steady pace, and then put in a strong forward stroke as you edge your kayak and try to lift your feet off the water. As the forward stroke ends and the bow leaves the water, turn the paddle movement into a reverse stroke to slice the bow down. As the bow enters the water keep your weight forward and keep the momentum going by rotating your shoulders to face the direction you are travelling in i.e. if you were edging your kayak on the right as you double pumped twist your shoulders to the right. As your body winds-up and then releases the kayak will follow. As you feel the bow lifting out of the water put in a forward stroke on the right hand side. Just before the kayak stalls and the bow comes over your head quickly twist again (to your right) to plant another reverse stroke in the water and throw the bow back down.
If your body rotation is good you can dispense with the forward strokes and depend solely on the reverse strokes. This will give the boat a 360 degree rotation on each and every paddle stroke making the cartwheel “clean”. This same technique can be used to throw cartwheels in the hole, spiltwheels, freewheels etc. etc.