I don't play Rekenton, so most of this will be of the somewhat generic nature. That said...
Play a champion that deals Magic damage. That hurts Rammus pretty hard as he'll have to stack Magic Resist instead of Armor (Rammus's Passive and W scale on Armor ).
Rammus relies on sticking with you for most of his damage. Jump in, slice/dice, and get out if possible, but that's gonna be difficult as Renekton or any other melee champion.
If playing AD vs Rammus (either counter-picked or just wanting to for some reason), get some help from teammates. Mages and other Magic damage dealers generally can do the trick.
Item build-wise, I can't exactly help much with Renekton specifics. Still, some of this may come in handy.
Build Magic Resist.
Rammus's abilities (Q, W, and R) and Thornmail all deal magic damage.
Damage options:
- 50 Magic Resist, 55 Damage, 10 Armor Penetration; Magic damage shield + 25% Attack speed, 10% Lifesteal and Spellvamp until out of combat
- 35 Magic Resist, 75 Damage, 10% Lifesteal; Taunt Cleanse + Burst of +50% Movement speed
Tanky options:
- 70 Magic Resist, 500 Health; Blocks first spell against you
- 70 Magic Resist, 500 Health, 10% Cooldown reduction; Increases all healing
- 60 Armor and Magic Resist, +20% Movement speed (near all structures); Helps push a lane or slow the enemy laner's push
- 60 Armor and Magic Resist; Get another chance to kill/escape every 5 minutes
- 25 Magic Resist, 30% Tenacity (Maxed Taunt goes from 2.25 seconds to 1.575 seconds)
Build Armor Penetration.
If he's gonna build tons of Armor, build some Armor Penetration.
- Mentioned above in Magic Resist
- 55 Damage, 300 Health, 20% Cooldown reduction; Cleaves Armor up to 30% when dealing physical damage
(Probably not optimal on Renekton) - 65 Damage, 10% Cooldown reduction, 20 flat Armor Penetration; 20% Movement speed + 40% Attack speed every 45 seconds
Build Health.
Health counters Attack, Magic, and True damage (not %Health damage, but Rammus has none in his kit).
Damage options:
- 400 Health, +25% Base Damage (14.5 - 27.75 Damage depending on your Level); When passive triggers, gain another +25% Base Damage + 30% Health shield every 45 seconds
(Probably not as optimal for Renekton) - 650 Health, 40 Damage; 40% Movement speed slow per auto-attack dealt
- Mentioned above in Armor Penetration
"Damage" options:
- 50 Armor, 500 Health; Gain stacks of movement speed, Deal Damage and a Slow when fully stacked
- 50 Armor, 500 Health; Deal AoE Magic damage to Rammus/enemies within melee range
Tanky options:
- 60 Armor, 500 Health; 15% Attack speed reduction when hit + AoE 35% Slow Movement speed every 60 seconds
- 850 Health; Gain substantial Health regen outside of combat
(Probably too niche for Renekton, though could prevent a kill if Rammus Taunts under his turret) - 300 Health, 50 Armor, 10% Cooldown reduction, +20% Movement speed (near all structures); Disable enemy turret every 2 minutes
- Mentioned above in Magic Resist
- Mentioned above in Magic Resist
Build Armor.
Rammus gains Attack damage equal to 25% of his Armor, so his auto-attacks get stronger the more Armor he builds.
"Damage" options:
- Mentioned above in Health
- Mentioned above in Health
(Probably not optimal unless stacking Armor/vs 3+ Attack damage champions) - 100 Armor; Returns a minimum of 25% of auto-attack damage as magic damage
Tanky options:
- Mentioned above in Health
- Mentioned above in Health
- Mentioned above in Magic Resist
- Mentioned above in Magic Resist