First, Dr. Mundo is a perfect hard carry jungler to climb with. He is great in 1v1s and is very tanky. He will win over most assassins with his ultimate and he has great ganks. I am currently sitting on a 93% winrate with Dr. Mundo in ranked, dominating every game with the following strategy.

You start by buying Emberknife and Refillable potion. In order to survive you need to walk around constantly to charge up your Fleet Footwork. You are not going to need any help/leash. Now, walk over to your raptors, upgrade your W and clear raptors by activating W, constantly moving, and only autoattacking the big raptor. After taking raptors you go over to red buff and upgrade your E, then you kill red buff while kiting it over to krugs. Remember to use your refillable potions. Smite krugs and kill them, then take Q and recall to buy a Rubycrystal. After that you walk down through the middle of the jungle and go over to the wall to the side of wolves then throw your Q in that direction. Now run as fast as you can over to your wolves and kill them. By throwing your Q at them they will get kited to the exact range thay can be kited if you do it correctly. Then you go to your blue, then gromp then see if the scuttle if taken, if not, ward the pixel brush (The small brush close to mid that is just a circle) and then take the scuttle. If any jungler comes you will for the most part be able to kill them but i don't recommend trying to. When it comes to ganks just try to land a Q and then walk in and try to kill, there is not much to it. Remember to constantly clear your jungle. When you get enough gold to buy a Bami's Cinder and a control ward, back and drop the control ward in the dragon pit and kill the dragon as soon as you see the enemy jungler appear top. If you just are level 5+ or above you wouldn't have any trouble killing drake. When you have your mythic and an item or 2 you should not at all be scared of tower-diving (Going under the turrets and trying to kill the enemy) as long as you have your ultimate and you are full health.

Since Dr. Mundo has such good clear you need to remember to always have your jungle cleared. If possible, try to take herald just remember to either use Oracle Lens or drop a control ward in the Herald pit before taking it.

If possible, try to give the kills to your teammates when ganking if they scale very good. (Kassadin, Twitch, Fiora)

When teamfighting don't be too scared as long as you have your items. I recommend always going for and early spirits visage because of the bonus heal to your passive and ultimate.

Thanks for reading and good luck with climbing.