Not Updated For Current Season
Cleanse is really powerful on Tryndamere right now. It can give you space to use your ult, or let you spin away from tough situations. The two second partial immunity after is all he needs to escape from most other champions.
Exhaust is wonderful too, in order to stop burst into you or a teammate since you don't have exceptionally high health, or to catch up to another champion. Frozen Mallet helps with in both regards, and combined with your spin and chicken shout, you can usually catch up and stay with a runner long enough to finish them off.
Armor penetration does more dps than stacking crit damage with this build, until about 90 armor. Most champions don't ever reach this value, and those who do usually take until endgame to do so. Not only that, armor penetration is substantially better early game (when Tryn is less potent), when armor is lower and IE has not been purchased. Also note that armor pen gets better as you get more of it, which is what makes Stark's Fervor so good. I use armor pen runes where possible, otherwise +crit/crit damage. The difference in dps is pretty close; as a general rule, if you have more than 55% (ish) crit with IE, +crit damage is better, otherwise use +crit. Although this build reaches about 90% crit, you spend a good deal of the game below that value (at least 25 mins or so before you get IE) where +crit is better. See here for a detailed discussion.
Formula for physical damage reduction: %reduction = ARMOR /(ARMOR + 100)
It's also perfectly viable (and more expensive) to stack full +crit damage, and will give you significantly better endgame damage. The problem with this is that Tryndamere's endgame damage already 3-shots non-tank heroes, so it's debateable whether it's worthwhile.
Only the 21 points in Offense are really necessary. I find that Tryndamere's resistances are low, so I'd throw the points in Defense, rather than the mostly useless Utility tree.
Normally I start with Brawlers and two health pots and upgrade to as many Avarice Blades as possible before 12 mins. Greaves are my second item. I then go Stark's, IE, PD and by then the game is usually over. I replace my Avarices only when my inventory slots are full. In order to make a profit, you need to have them for at least 12.5 mins. Even without turning a profit, each one grants you 12% crit, which is excellent.
Lifesteal is counter to Tryndamere's design; you gain crit as your health gets lower. However, it greatly helps survivability in small fights. Try to find the optimum amount for your playstyle/team.
Stark's Fervor: If your team already has one, although you can have 2 stacks of the buff (theirs and yours), those without Stark's will only get one.
Executioner's Calling: Really good if the other team is stacking lifesteal (Warwick, Nasus, etc), or against a healing support unit.
Bloodthirster: Excellent item if your lifesteal is below about 20%
Last Whisper: This is only really better than the other items if you're team doesn't benefit from Stark's and you don't want lifesteal. The 40% armor reduction is calculated after Stark's, but before your flat reduction.
Black Cleaver: Once you have high attack speed, this is just about the best dps weapon you can get other than IE.
Guardian Angel: If you're getting focused (and dying) often, Guardian Angel is a great defensive item.
Frozen Mallet: An alternative to using Guardian Angel with a slowing and dps component. If you're never focused (not likely), it's not worth the gold. It would still be worth upgrading to Phage, with a 40% chance to slow.