I prefer a splash-damage heavy Kayle jungle, so I usually go Sorc Treads(or defensive boots depending on enemy comp), Elder Lizard, Nashor's Tooth and Liandry's as core items, and after that I either aim for tanky stuff or more AP(Deathcap, Zhonya's, Rylai's)
But the Elder Lizard, Nashor's and Liandry's are definately core, because they all make your Q stronger(and Q makes Liandry's stronger on that target) and both of the DoTs work with your E.
For skill order I start E(obviously), then go either W or Q, depending on how much help I got on dogs/blue, and at 3 I take the ability I haven't put points into yet. After that I max R>Q>E>W. The reason for taking W before the game forces you to is for the sprint, which helps quite a bit.
For masteries I go with my special Kayle page(21/9/0) getting both types of penetration in offense and sticking the rest in the defensive masteries that everybody takes.
AS reds, MS quints, flat armor yellows, scaling MR blues.