I'm definitely not the best Jungler but here is how I see it. And I may repeat some things that you already know:
So Counter-Jungling, specifically, refers to entering the other team's jungle in an attempt to either 1) steal their camps or 2) kill the jungler himself. Counter Jungling relies on quite a few factors - your champ, the other Junglers champ, what side of the jungle you are on, and how well you can communicate with your team.
Your champ: Some champs are better at counter-jungling than others. If you're going to steal camps, you want to be able to get in and out of their jungle easily and clear camps fast, so you can steal them before you are found or their jungler arrives. Tank junglers tends to have slower clears and aren't usually good candidates, and its also easier to steal from them (since, without smite, they usually have lower damage abilities). There are some exceptions; Rek'Sai is a tanky jungler, but her tunnels allows her extreme mobility throughout the jungle and makes it very easy to get to camps without being seen and avoid getting ganked, and her clear isn't too bad if you pick up a little AD.
Their Champ: Some enemy junglers are better to counter than others. Nunu is generally someone you do not want to counter gank - he can stay at full health throughout his clears by Consuming the jungle camps, and he has a very strong targeted slow, and a self-buff that buffs his movement speed, so you really can't catch him in the jungle 1v1. If Nunu is in the game, chances are he will be stealing your camps.
If you're going to try to kill their jungler, you usually want to hit them early before they have enough sustain to get through their camps without being low health. If you can ambush them early on, you want to hit them hard and fast, kill them, and get out. The reason is that as soon as you are spotted in the jungle, they will likely ping for support and their allies will likely come running and you'll quickly be outnumbered and die.
If you're going for camps, the Tracker's item gives you a bunch of extra wards that you can use to scout out their camps and see when they come up so that you can get there quick and steal. However if you're going for a kill, the Skirmisher's is usually better since its Smite makes you take less damage from 1 champ and deals true damage on-hit.
So there are some things to think about. Another big part about jungling - whether you're countering camps, going for a gank, or looking to hunt their jungler, you want to watch your map closely. If you see the enemy jungler going Top for a gank, head to their bottom side jungle and clear out their camps. Or you can take the opportunity to gank bot, since you know their jungler won't show up. As you pass through your own jungle, drop wards in the river, somewhere where you can see the Jungler if they go to 1) gank a lane or 2) kill the Rift Scuttler. Killing the Scuttler yourself is also a good choice, since it gives a large vision area.
Anyway, the basic idea is that when you see the jungler, go steal whatever camps they are furthest from. They are more likely to have spawned again in the time it took them to clear the rest of the route.