Ok lets break this down for you real simple like...
1: Summoner Spells.
A: Cleanse. No, no arguing. A CC to Tryndamere is like having all your spells locked down as a caster for 5 seconds. A focused Tryndamere is still a problem. A CC'd Tryndamere is a dead one.
B: Ghost, Ignite, Smite or Exhaust. Ghost is the direct choice. Get exhaust and ignite to deal with gimmick teams (IE: double support, Carry-vile, ect) Smite is for jungling obviously. Flash is for mouth breathers that can't be arsed to learn what positioning is. You already have whirling blade- if you can't escape with that AND ghost you weren't meant to live. If you can't get a kill with the two of them AND mocking shout, you weren't meant to get it.
2: Masteries: 21 : 9 : 0
You want just about everything physically oriented for tryndamere in the offense tree. Take CDR over attack speed- you have many options for one of the two, but basically only one for the other. Nine into defense for Nimbleness.
Reds: Armor pen. Crit trumps this when you KNOW there isn't going to be much breathing room.
Yellows: Dodge. No, no excuses.
Blue: CDR is fine. Otherwise Magic Resist.
Metas: Armor Pen. HP (Flat, or per level) works too if you're the lazy type and want general rune builds.
Generally: 2 items with crit, at least one significant life steal dump. Two open slots, and boots.
Too much crit makes your passive useless, and infringes on the whole **** point to your ult (no, it's not JUST about staying alive.) Without Lifesteal you're just useless.
Your core item build is always going to be Boots + infinity edge + Phantom Dancer. That's 65% crit without factoring crit scaled to HP, or runes and masteries.
Infinity Edge is hands down the single most important item for Tryndamere. No arguing. Single largest damage improvement for him. Only thing you can reasonably prioritize over it is your initial brawler's gloves and level 1 boots. You don't want a phantom dancer over it first- in terms of high price marked items AD always trumps attack speed.
"Gee Willikers Acosn, why don't I just stack phantom dancers rather than getting boots?"
For one, it's too expensive. Tryndamere already has a fairly deep-set "carry" period where he can start doing his job. Throwing in 2-3 phantom dancers pushes that back further. Plus, if your head's on straight you'll remember that merc treads (because cleanse is on CD sometimes, and you're never going to see a team with less than three CCs) are more than just movement speed.
"What about your two open slots?"
If you need to deal with tanks grab a last whisperer, or madred's blood razor. If you just want raw damage blood thirsters and / or a black cleaver. Youmuu's is a natural choice for any build because it's literally everything you want.
"What about a QSS?"
What about it? It's useful, fairly inexpensive too. You should only really need it for teams that get exhaust / ignite happy. Yes, it removes those. The only flaw with it is that it's only magic resist, and it's on-use. Merc treads does 3/4ths of that and has movement speed too, and you're getting cleanse anyways.
"What about Lifesteal?"
Short Answer? Blood Thirster. Need more? Another blood thirster.
Long answer: Blood Thirsters are useful so long as you can consistently stay alive. Being realistic there's teams that are just going to be arsed enough to hunt you. There's always another eve hiding for you, and sometimes Karthus just hates you more than the rest of your team. In that case, Stark's is a strong alternative. It cuts enemy armor, and gives attack speed too. Doing half it's effect as an aura doesn't hurt either.