Trinket is personal preference. I like to start ward trinket then maybe swap for red later in game to make picks.

Rushing wriggles will help your jungle clear, sustain and gold.

Bers or Sorc boots after that... Sorc boots a better option if you want some early MPen for your ganks, Bers for farming.

Next start constructing your Nashor's Tooth. Get the AS parts first. This item is made for Teemo. Helps you jungle and gank better than rushing Haunting Guise, especially as you are maxing your E first.

Get Liandry's next if the enemy team are being hit by your shrooms often, otherwise get it later. Ff enemies are not building health, Liandry's isn't so good either.

After that, get Runaan's Hurricane... Yet another item made for Teemo!!! This will allow you to kick butt in team fights, as you will be quickly spraying out AOE Toxic Shots.

Get haunting guise here if you haven't already.

Throw in a Deathcap next to make you even more OP.

Your last item (after selling wriggles) is personal preference. If you are getting burnt in team fights, get a Zhonya's. If you want even more damage, get a Lich Bane.

Maxing Toxic Shot first gives you the best clear. First point must be spent in Blinding Dart however, as this allows you to tank a large part of your smiteless leash almost unharmed. Once you have a point in Blinding Dart and a point in Toxic Shot, your jungle sustain should be fine and you will in theory be spending most of your time ganking anyway. Maxing Blinding Dart first is only viable if you are against VERY mobile and hard to gank laners. You only need to get in 3 auto attacks before your Toxic Shot points pay off.

Armour Seals and AS Marks are the only really important runes for Jungle Teemo.

I have filled the rest with hybrid damage... Feel free to experiment and comment!

You can get MS Quints and first point in W if you feel like a cheeky buff steal to start the game.

There should be no such thing as a specified "Jungle Route". You go where you think you should go at the time you think you should go there. Use your head. This is one of the things that makes Teemo a viable jungler. Unless the enemy jungler is a really useless duelist they will almost definitely try to CJ you... Anticipate when and where and get a laner to come kill them. Sub-par duelist junglers can be some of the best counter junglers in the game if played right, as you are not expected to be CJing and can expect to be CJd yourself. Use this knowledge to your advantage. Just don't try anything too risky.

Simple Start: Smiteless red with a team mate watching your blue for you. Take a small camp, go kill blue and then start ganking ASAP. Look at the inventory of all enemy players to figure out who has wards and where they are. Ask team mates about enemy ward placement before you start moving to a lane... Don't want to waste a second of your time.

Whenever you see the enemy jungler, use that information. Unless its a waste of time, try to counter gank him if you are close, or steal some jungle if far away. If he goes into his jungle low and you think you can kill him, do so. If you ever see him without a buff when he should have one, get a buddy to distract him while you go take it, or organise a little surprise party at his buff camp.

A good ganking trick is to tell a lane to push, go stand in it and invis then ask them to quickly let the lane drop. Pop out of invis and you have a kill. Not so easy in bot if they are warding through lane.

You have invis and smite. LOLOLOL... Reds, Blues, Dragons, Barons... They are yours.