Why Conqueror instead of Lethal Tempo?
Both runes are for longer fights, but Conqueror has more value in any stage of game, because it provides bonus AD and you Heal for 8% of the post-mitigation damage from damage dealt to enemy champions. Lethal Tempo otherwise has low use, because you don't have limit for attack speed and extra Range is useless for melee champions.

Triumph is the only rune in first line that works with bel'veth, as it Heals for 10% of your max life and kill/assists gives you 20g. Overheal works, but nobody uses it. Legend: Alacrity is worthless as it add just a bit of attack speed on top of your infinite stack. Legend: Bloodline give 100 extra HP and 6% lifesteal which is great as you'll have better sustain in-fights. 3rd line is pure personal prefference, as Coup de Grace synergyses well with Royal Maelstrom to finish people, and Last Stand works well with your Royal Maelstrom's lifesteal and Immortal Shieldbow.

Overgrowth is a solid option if you want to stack Health, along side with Bone Plating or Conditioning. Conditioning has more value throught the game as it scale with your resistences. It's a great scaling rune, turning Bel'veth in to a extremely resilient bruiser.

Future's Market is a nice option to rush your core items, and get a little bit of gold advantage. Magical Footwear gives some nice mobility, but it's a bit riscky to delay your boots.

Sudden Impact has absurd synergy, but lethality is a really weak stat right now. Zombie Ward otherwise is extremely strong rune, as you negate and replace vision, also gets stronger higher the elo. Eyeball Collection is inferior, only take that if you're a low elo. Relentless Hunter gives more out of combat speed, which synergizes well with Bel'veth's true form passive.

Flash gives you some extra mobility.
Ignite gives you dueling power and kill presure.
Smite obligatory.
Challenging Smite or Chilling Smite?
Challenging Smite is great for duel power, as it take away 20% of your enemies damage. Chilling Smite makes you EXTREMELY sticky, making almost impossible to escape once you get close.

Core:
Immortal Shieldbow gives everything Bel'veth want, AD, attack speed and lifesteal. Blade of ruined king is the same, but you steal 20% of your target speed and deal damage based on your enemy's life.
Situational:
Maw of Malmortius and Wit's End both for MR, but the first one, you build if you're getting IK and the other for some resistance and extra damage. Death's Dance is a solid option, as you'll get some armor, damage and healing. Bloodthirster gives lots of lifesteal, which is great with your core items, besides the extra crit. Guinsoo's Rageblade is an overhyped item, as it's real value come to a full crit build, and Death in Lavender nerf Guinsoo extra damage. Both Hydras are good, you can choose Ravenous Hydra for insane lifesteal or Titanic Hydra for extreme resistence.

BIG THANKS for reading until now and if you have any suggestion of what to improve in this guide, feel free to comment, I'll surelly read. With time I'll make this guide better. Check out my Support AD Shac
AD Support Shaco guide!