Every champion has its own builds, it really depents on if you are an ability power champion, an attack damage champion, or a tank.
Every champion starts with boots, potions or a starting item which increases your damage.

An ability power build for Karthus (APCary):

• Starting items.
  Mana Potion:
  Restores 100 mana over 15 seconds.
  
  Health Potion:
  Restores 100 Health over 15 seconds.
  
  Sapphire Crystal:
  + 200 mana.
  
  Warding Totem:
  Places a Stealth ward for 60 seconds.
• Example Finished build.
  Righteous Glory:
  + 500 health, + 300 mana, + 60% movement speed when moving towards an enemy.
  
  Sorcerer's Shoes - Alacrity:
  + 15 magic penetration, + 15 movement speed. Alacrity +20 movement speed.
  
  Archangel's Staff:
  + 80 ability power, + 250 mana. Gain ability power equal to 3% of your maximum mana.
  
  Randuin's Omen:
  + 60 armor, + 400 health. If you are hit by a basic attack, you slow your attacker.
  
  Thornmail:
  + 100 armor. When being hit by basic attacks, return 15% of the damage taken.
  
  Abyssal Scepter:
  + 70 ability power, + 50 magic resistance.

An Attack damage build for Ezreal (ADCarry):

• Starting items.
  Doran's Blade:
  + 7 attack damage, + 70 health, + 3 life steal.
  
  Health Potion:
  Restores 100 Health over 15 seconds.
  
  Warding Totem:
  Places a Stealth ward for 60 seconds.
• Example finished Build.
  Trinity Force:
  + 30 AP, + 30 AD, + 30 AS, +250 health, + 200 mana.
  
  Muramana:
  + 20 attack damage, + 1000 mana.
  
  Ionian Boots Of Lucidity - Alacrity:
  + 45 movement speed, + 15 cooldown reduction. Alacrity + 10 movement speed.
  
  Last Whisper:
  + 40 attack damage, physical damage ignores 35% of targets armor.
  
  The Bloodthirster:
  + 80 attack damage, absorbs 50 - 350 damage.
  
  Mercurial Scimitar:
  + 80 attack damage, + 35 magic resistance.

A Support build for Nami (Support):

• Starting items.
  Spellthief's Edge:
  + 5 ability power, + 2 gold generation. Spells and basic attacks against enemies deal 10 additional damage and grant 5 gold.
  
  Health Potion:
  Restores 100 Health over 15 seconds.
  
  Mana Potion:
  Restores 100 mana over 15 seconds.
  
  Warding Totem:
  Places a Stealth ward for 60 seconds.
• Example Finished Build.
  Boots Of Mobility - Homeguard:
  + 25 movement speed. Homeguard + extra fast regen of health and mana.
  
  Locket Of The Iron Solari:
  + 10 cooldown reduction, + 400 health, + 20 magic resistance.
  
  Frozen Heart:
  + 90 armor, + 20 cooldown reduction, + 400 mana.
  
  Oracle's Lens:
  Disables and reveals nearby hidden wards and traps for 6 seconds.
  
  Zeke's Harbringer:
  + 50 ability power, + 30 armor, + 10 cooldown reduction, + 250 mana.
  
  Mikeal's Crucible:
  + 10 cooldown reduction, + 40 magic restistance. + 100% base mana regeneration.

A Jungler build for Amumu (Jungle):

• Starting items.
  Hunter's Machete:
  Deal 30 magic damage on hit to monsters over 2 seconds and gain 7 health and 4 mana per second.
  
  Health Potion:
  Restores 100 Health over 15 seconds.
  
  Sweeping Lens:
  Disables and reveals nearby hidden wards and traps for 4 seconds.
• Example Finished Build.
  Boots Of Mobility - Homeguard:
  + 25 movement speed. Homeguard + extra fast regen of health and mana.
  
  Stalker's Blade - Cinderhulk:
  + 400 health, + 15 health multiplier. Deal 45 magic damage to monsters.
  
  Dead Man's Plate:
  + 50 armor, + 600 health. You build up stacks, increasing movement speed.
  
  Liandry's Torment:
  + 80 ability power, + 300 health. + 15 magic penetration.
  
  Spirit Visage:
  + 10 cooldown Reduction, + 400 health, + 60 magic restistance. + 150 base health regeneration.
  
  Frozen Heart:
  + 90 armor, + 20 cooldown reduction. Reduces the attack speed of nearby enemies.

A Tank build for Rengar (Tank):

• Starting items.
  Doran's Shield:
  + 80 health, + 6 health regen every 5 seconds. Blocks 8 damage from enemy basic attacks.
  
  Health Potion:
  Restores 100 health over 15 seconds.
  
  Warding Totem:
  Places a Stealth ward for 60 seconds.
• Example Finished Build.
  Mercury's Treads:
  + 25 magic penetration, + 45 movement speed.
  
  Sunfire Cape:
  + 450 health, + 45 armor. Deals 25 character level damage per second.
  
  Spirit Visage:
  + 10 cooldown Reduction, + 400 health, + 60 magic restistance.
  
  Randuin's Omen:
  + 60 armor, + 400 health. If you are hit by a basic attack, you slow your attacker.
  
  Ravenous Hydra:
  + 75 attack damage, + 15 health regeneration per 5 seconds, + 12% life steal.
  
  Bonetooth Necklace:
  Rengar collects trophies when killing champions and gains bonus effects based on how many trophies he has.