Shen's place in a team is still debatable in consideration as to where he belongs in either in top lane or in the jungle. Both places work well and both places have strengths and weaknesses that can be exposed. Regardless of where Shen lanes he will still be able to transition into the same roles and duties. This article isn't to say he's a better jungler than laner but instead it is merely to go in depth into what Jungle Shen is. Ultimately it is still up for debate as to where he is best.
Jungle Shen, or just Shen in general, has been one of the top picks or bans since he was buffed awhile back. That buff made his jungling much stronger and effective and made him a viable choice in the jungle. He's currently one of the best jungle picks in the game in the sense that he enables your team to have a strong fighter and global presence. His transition is also a threat to the enemy team as Shen alone will force the enemy team to change their playstyle once team fights occur.
Anyways let's break it down shall we?
Speed - Shen is arguably one of the slowest junglers in the game and his speed doesn't actually even pick up. Normally you wouldn't be buying any items that speed him up either (Recurve Bow comes after your gold items) so his speed remains poor. This leaves him susceptible to being invaded by other junglers. He can deal with duelists decently (especially if double buffed) but he can't really finish them off. He has severe issues dealing with thieves such as Shyvana and Dr Mundo so if he gets no assistance from his team in the form of back up or wards then consider him boned. His slow speed is no issue vs ganking junglers such as Nautilus and Maokai who aren't looking to fight you in your jungle anyways.
Ganking - Shen's pre 6 ganking is rather strong. His damage output is rather good and dash is both a strong gap closer and crowd control. He is able to bypass certain ward spots and walls due to his dash being able to wall hop. His level 2 ganks can be risky since if he fails he sets himself behind quite a bit. However his level 2 ganks are better than most (once more because of the nature of his dash) especially if he did a red start. A double buff Shen unleashes incredible amounts of control on his target often being able to dash twice during the gank whilst applying the slow and dealing great amounts of damage. He is a jungler that is gold dependent and slow so he will need to get ganks in order to make up for that weakness.
Once Shen reaches level 6 he obtains what is called "infinite jungling" and opens up a unique jungling power only known to him. His ultimate is a global teleport that shields his teammates. That alone should let you know what infinite might be hinting at. In short since he has a global ultimate he is able to jungle or split push all day without losing the ability to pressure the enemy team as long as his ultimate is up. If the enemy team tries to gank his teammates he can teleport in regardless of where he is or what he's farming and counter gank. If his teammates go for a gank he can teleport in to assist the gank. Shen may not even have to use his teleport to pressure the enemy team. The mere fact that he can show up at a moment's notice means the enemy team will think twice before committing to anything. This in the end gives Shen "artificial" speed as the enemy jungler still has to roam the map in order to do things while Shen is using that time to farm non stop.
Match Ups - As it was stated earlier Shen will lose to fast junglers. He's a very risky pick vs junglers like Udyr, Dr Mundo and Shyvana especially if his teammates aren't pushers and can't pressure their own lanes and come to his assistance. Match up wise he has some of the poorest match ups in the game. He is either even or negative with most. Junglers that gank lanes don't bother him much but some can still bully him if they want thanks to his atrocious speed. When it comes to dueling Shen is pretty strong at 1v1s but not strong enough to kill or survive a duel with the junglers known for dueling. His best bet in most cases is to flee which ends up costing him a camp and time. This is why he resorts to ganking the laners which he has better match ups against. He can gank most laners and hope his teammates are able to mop up the kill.
Builds - Rune and mastery wise he's pretty standard. He is able to rune armor yellow only and do well. He doesn't have much in terms of rune commitment. For masteries just straight down the defensive tree and the other 9 points are free to go into offense or utility. I personally prefer utility to grab the movement speed, energy and buff duration. As for Runes I personally go Attack Speed Red, Armor Yellow, Magic Resistance Blues and Movement Speed Quints.
He can jungle naked and that is to say he can jungle with any item he wants. He can jungle with doran's shield, ruby crystal or even amp tome. All those are gimmicky of course so often times you'll find yourself going boots + 3 health potions. This will help you gank and move around the map easily clearing camps as you go. Thanks to your runes and masteries you'll likely be the fastest player in the game early on.
Shen can opt for a multitude of builds. He can be a fighter with wit's end + offensive items or he can be an aura support or full on tank. He can survive solely on gold items and transition into whatever he wants. He can change his build on the fly to suit the team's requirements. I have gone into a game fully expecting to go full tank Shen but was able to get a good sum of gold early that I changed into a fighter and became a huge damage dealer for my team and nuisance for the opposition. Shen's only guaranteed item is boots and everything else is up to the whims of the game and the needs of the team.
I would recommend using Shen even if he may countered simply because of how useful of a champion he is. However I will say he's a gamble so hopefully your teammates are on the ball for this. On a more "competitive" level he is best used when your team can force the risk on the opposition. That means while Shen is a risky pick and vulnerable to being counter picked and invaded if your teammates pick champions that can push and gank and thus would make it more difficult for the enemy team to invade then you force the risk onto them. If they try to invade and do it poorly it can end up being catastrophic for them. You will be able to dictate their picks as well. If they see Shen picked they will either have to lock in strong pushers/laners early leaving themselves open to counter pick or they will have to pick a team that can deal with his split pushing later in the game.
So in conclusion - Shen is an A-Rank jungler that adds a lot of unique powers to a team. Picking him gives your team a lot of room to shape the team comp however you want so consider him for a jungler if you opt for a pushing strategy or a gank heavy strategy.