This build is not for beginner Tryndamere players. You need to know his passives and abilities very well or else you will just get reported for troll and feeding. At early levels you are not as strong as AD Tryndamere and as such you will have to play passively. The key to this build is farm farm farm farm. You need to get gold as fast as you can because you will be completely dependent on your items. Once you get your Nashor's Tooth you will see what I mean. When you add your Sorcerer's Boots and Lich Bane to the equation you will be able to be more aggressive and your trades will always be in your favor. In end game, after you have gotten your Rabadon's Deathcap and Void Staff, you will be able to nearly completely heal your max HP with full fury stacks and then toss in the Zhonya's Hourglass for the AP boost, Def Boost, and the 2 sec of being invulnerable will give you that slight edge to be able to pop your Bloodfury heal again. The only thing that will be able to stop you (which is the case in AP Tryndamere as well) is CC. If your enemy team has no CC, then you will be able to 1 v 3 them and in some cases even be able to 1 v 5 them. If you do it right, you will be laughing and wondering why they continue to keep throwing themselves under your sword.

The main reason why you will need Lich Bane and Nashor's Tooth is not only for the AP and utility they provide (LB gives 5% movement speed and NT gives 50% attack speed and 20% cooldown), but it is their passive that really give them the edge.

First off, Nashor's Tooth:
Your auto attacks are going to lose the normal "hit and kill" feel that comes with playing AD Tryndamere (more specifically Crit Tryndamere) and Nashor's Tooth will give that edge back periodically. NT's unique passive adds 15+15% of your AP to your autos, and when you piggy back that with your mastery "Spellsword" will give you another 5% AP to your auto attacks. Once you reach full build and you have 680 AP, each auto attack will add 151 bonus magic damage to your auto attacks.

Second, Lich Bane:
Lich Bane is the hidden "kill switch" for AP champs. When you use an ability your next auto attack will do 75% your AP in bonus damage. That gives you 510 bonus damage on your auto attack once you reach full build. This passive will prot with each ability on your roster (even your Bloodlust to heal)

With these two items it is simple: Spinning Slash in (870 damage) Lich Bane prot (510 bonus damage) and auto attack (151 bonus damage). So far that is 1531 damage in bonus damage alone. Then you activate your "Mocking Shout" and prot Lich Bane again and auto again for another 661 bonus damage and they are slowed. If they are still alive you can just auto them down until "Spinning Slash" is back up and boom, you will kill just about any other champ.

Also, if they happen to fight back and widdle your HP down, you can activate "Bloodlust" to heal yourself for around 1500 and also prot Lich Bane for another hard hit. And if you are fighting more than one enemy and are getting widdle again, pop "Undying Rage", Lich Bane prot, and just before it ends activate "Bloodlust" again to heal and Lich Bane prot again. Damn near unkillable...