I mained Trynd for a while and have about 120 games with him under my belt, I think.
Great damage output
Extremely high mobility
Great squishy killer
Fun champion to play
Has difficulties dealing with exceptionally fat champions
Cannot deal with CC
Extremely farm and item dependant
Requires a high degree of skill to play effectively
Spinning Slash is buggy.
His design is inherently flawed for 5v5 combat.
Presently, I would not recommend buying him. He does not fit well with the current metagame at all. I would dare say that there is not a single instance where Tryndamere would be an appropriate/good pick. Master Yi will almost always be a much better champion if you need a carry.
This is due to Tryndamere being sort of flawed as a champion. He relies to heavily on abusing his passive for damage, much like Olaf, but has no way to keep himself alive beyond his ult. His ult, rather than directly dealing damage, serves to keep him alive at his most dangerous. However, since he's so easily locked down during a fight, he is rarely able to effectively deal much real damage.
The problem is exasperated the further you climb in Elo. Better opponents will be able to burst Tryndamere down before he can react and use his ult, or chain their CC to prevent him from attacking during his ult. Further, it becomes increasingly difficult to live after using Tryndamere's ult, since a single errant ignite or DoT can kill him as Undying Rage comes off cooldown.
Quicksilver Sash can help as well, but Tryndamere is so incredibly dependent on farming items to have competitive damage that the cost is seriously detrimental early game to his damage output. Unfortunately, he needs it early/mid game to excel and survive late game. It's something of a catch-22.
This is not to say Tryndamere can never be effective. I have had success with him on TT, where the fights tend to be much smaller and his mobility is very valuable. If Tryndamere can fight small scale battles, he will win versus most other champions, such is his prowess in 1v1 or 1v2 situations.
If the metagame shifts toward something resembling the EU, Trynd may have a place. His high mobility and ability to excel in small scale engagement would offset his weaknesses in team fights and might allow him to be played as an incredibly effective assassin type champion. He still may be overshadowed by the likes of Shaco in that regard, but he would have a more appropriate place and may be effective.
tl;dr He's fun, but not effective. I would not buy him if you're looking for a serious champion to play.