From my experience at jungle with tryndamere, the summoner spells i find most useful are smite and ignite. The reason goes as follows, smite is obviously easier to jungle with, and ignite is also a good jungling tool along with being useful to kill enemy champions later on in the game. Exhaust is a waste with trynd since his Mocking Shout (W) automatically slows the enemy down. Ghost is also a waste since having boots and phantom dancer pretty much makes you faster than anyone else in the game. Flash is just a no, your spinning slash (E) is your flash. Heal, idk why u would get heal for trynd, he has his Q and infinity rage, if u need heal to survive then u shouldn't be playing trynd. Lets see the other summoner spells are clarity and clairvoyance which are pretty much useless for trynd. The tower protect spell (the name has slipped my mind) should be used by your support.
Now that i got the spells out of the way, the items i go with is cloth armor and 4 red potions. You can get 5 but you really dont need the 5th one unless you want to be extra catious. The first thing i do when i jungle is get red lizard with the help of a teamate who should leash it. As long as you get one crit hit on lizard plus the smite and ignite, you should kill it with about 50 health or so. You will need to use a potion imediately after the first hit otherwise you will not survive. The best leashes are when your teamate hit the lizard 3-5 times. This way you will still have half your health and your teamate will also have about 3/4 of their health remaining.
Immediately after lizard i get both golems. Attacking them while using another potion for healing. Smite will still be on cooldown and should not be used while fighting these 2 golems. While fighting the golems it is helpful to hit and slightly run away and occasionaly hit the same golem until the second retreats. This way you essentially only take the damage of one of the golems. Once smite reactivates, take out the wraiths. Spin slash into the wraiths while simultaneously using smite on the blue one. You should easily kill them while gaining health so dont use a potion here. Go to the wolves and kill. Now you can take blue golem. Notice, you will risk dying so do this if you feel comfortable with your trynd skills. Start off by killing both of the minions. Then start attacking the golem. Whenever your rage hits 100 heal. Your smite should reactivate mid fight with golem so use it. From there determine whether you can kill the golem or have to retreat. I end up killing the golem half the time with less than 100 health remaining. Now you can go back to base and get your wrigglers lanterns and a pair of boots.
From here on out i am pretty sure you can figure out what to do.
The order to rank up your abilities are: Q-E-Q-W-Q-R (Then max out your Q and W, with E being maxed out last)
If you have a decsent team then you will easily destroy the opposition since you will be about 1 or 2 levels ahead of everyone.