At the start:
Doran's Blade: Standard. Gives a lot of stats for only 475 gold.
Long Sword and 2 Health Potions: The alternative if your support has no sustain in their kit.
6x
Elixir of Fortitude and 6 Health Potions: Another alternative. Good for lanes who you want to trade a lot against, or if you know the enemies are aggressive.
First back buys:
Ideally, you want this to be the item you buy when you recall for the first time. Will build into Infinity Edge or Bloodthirster.
Another flat attack damage item you could buy if you couldn't but the B.F. Sword. Builds into Infinity Edge, Last Whisper and Bilgewater Cutlass.
Vampiric Scepter will grant you lifesteal which return health on auto attacks. This lets you stay in lane longer. Builds into Bloodthirster.
Another item you can buy if you have leftover gold. Builds into your Berserker's Greaves.
Like Boots of Speed, for the extra gold you have. Builds into Zeal, Phantom Dancer, Berserker's Greaves, and Blade of the Ruined King.
Possibly? More gold I guess. Builds into Statikk Shiv.
Mid game:
Boots of choice. You will enchant this with either furor, homeguard, or alacrity.
Zeal gives you attack speed, movement speed, and critical strike chance. Builds into Phantom Dancer or Statikk Shiv.
You'll take either this item, Statikk Shiv, or Blade of the Ruined King. Phantom Dancer gives you more attack speed and critical strike chance. Better for single target DPS.
Statikk Shive provides an extra 1% movement speed and can deal magic damage to 4 nearby enemies. It provides less attack speed and critical strike chance, though.
Blade of the Ruined King grants life steal and attack damage, but does not grant movement speed or critical strike chance. It also deals % of and enemy's health per autoattack. It's active is great because it steals their life and movement speed.
Infinity Edge grants flat attack damage, critical strike chance, and makes critical strikes do more damage than normal. Does not grant lifesteal like Bloodthirster does. Grants less attack damage than a stacked Bloodthirster. Build this on champions that don't have attack damage scaling on their abilities because they're auto attacks are going to be their only source of damage.
The Bloodthirster grants lots of attack damage and lifesteal. Build this if you want the lifesteal over the critical strike chance on Infinity Edge.
End game:
Last Whisper allows you to ignore some of the armor your opponent builds. Also grants some flat attack damage.
Attack damage carries will build only 1 defensive item. Warmog's Armor will not do you wrong. It gives a huge chunk of health.
Frozen Mallet grants less health than a Warmog's Armor, but also grants attacks damage and can slow enemies for kiting.
Guardian Angel will revive you after you die.
Mercurial Scimitar grants magic resist, attack damage and will remove crowd control. Build a Quick Silver Sash since it builds into this.
Highest armor item. Build this if your opponents have tons of physical damage.
Randiun's Omen is like Thornmail, but will slow your enemies and decrease their attack speed. Grants health. Probably better than Thornmail if you want to counter physical damage characters.
Zephyr will replace your boots if it comes to a 6 item game. Grants attack damage, attack speed, cooldown reduction, and most importantly is the tenacity.
ERMAGHERD THIS GAME NEVER ENDS!!!!!!:
Just buy an array of these if it comes to this point in the game.