I honestly think Udyr is one of those favoured by the new (crappy) meta. He does better in S10 than he did in S9 because he can slap the enemy jungler from the get go. You shouldn't really be splitpushing tho, you don't have time for that. Even counterjungling is only worth it if you steal buffs and gromp or kill the enemy jungler in the process.
Your priority list should look something like:
free gank / countergank > scuttle > invade > dragon > regular gank.
Keep in mind Dragon is very important but you should only prioritise it based on vision, lane priority and info on the enemy jungler (aka sneak it or take it with botlane)
Start with a rejuvenation bead and talisman, first clear Buff-buff-gromp then invade enemy at his second buff, steal the buff, kill him (or force his flash) take scuttle, transition/gank mid, take second scuttle, reset, rush Tiamat. This should be your first clear ideally. From there you can keep on invading or taking easy ganks while you control Dragons and scuttles. You try to always take Krugs and Gromp on your way to and from your objectives but you can ignore raptors and wolves when you don't have time.
I had the most success getting trinity force after tiamat, I then finish titanic and cinderhulk to then focus on armor and hp, I'll always take wit's end if I need MR. If the game looks short and snowbally you can do Warrior - Ravenous Hydra instead, very strong burst early, doesn't scale as good late because it's squishy, fit a recurve bow in there, then you can round it up with Frozen heart and GA. If you need MR you'll always do wit's end, if not you will close a Rageblade at the end. We play with conqueror and Legend: tenacity so we buy ninja tabi almost every time.