The core of CP Adagio contains two items: Alternating Current and Broken Myth. The third item should be Frostburn if you have to kite/chase enemy or Shatterglass if not. For defences you shuld buy Atlas Pauldron if playing against WP melee hero (but not lance) or Metal Jacket if not. There are two viable boots for Adagio: Journey Boots and War Treads. War treads make your Gift of Fire heal stronger, but you are slower, and Journey boots are usefull for kiting.

Adagio is weak against ganks. If you are plaing against Glaive, Koshka or similar classes you have to upgrade your Gift of Fire faster than Agent of Wrath (up to 3 on the 4 lvl). Getting lvl 2 boots just after Alternating Current is useful on the lane too. If you get ganked by any class you should first use A on yourself (it slows enemies around you), use boots and try to get near your turret. Buying Frostburn helps too. you can easily kill them if you block their stuns, heal youtself, throw Agent of Wrath on yourself and melt them before the melt you.

Adagio CP Lane use his Gift of Fire on his teammate who is close to enemy to burn them. then he uses Agent of Wrath on himself and melt his burning opponents. When his enemy is burning, adagio can solo nearly all heroes so dont get scary. Be aware of Atlas Pauldron - it shuts you down.

Use your ult very carefully and only if there are 2-3 of them near you. First, splash your arcane fire on your enemies by Gift of Fire. Then use Verse of Judgement. Activate reflex block if you there are some enemies with stun (Catherine, Koshka, Glaive, Joule, Lance,Finn lvl 8).

If plaing jungle, you should get in the middle of team fight,and then use Verse of Judgement as writed above. Well placed and not responded Adagio's Ult is winned teamfight, but if not, it would waste time that can be used by oppenent.

!NEVER USE YOUR ULT ON NOT BURNING OPPONENTS!.

Arcane Renewal. When playing Lane you should place half of your Gift of Fire on lane minions or roam to burn enemy lane minions to get energy back to you. When playing in the Jungle just place Your A on yourself.

Adagio CP Lane is best with Blackfeather WP, Krul WP and Lance WP. He is good with any melee hero. Becouse Adagio can be played on any position you can pick him first in drafts. if enemy find counter against him, you can play roam adagio instead of CP lane/jungle.
How Powerfull is CP Adagio Lane? see that videos:
Cloud9 wins VISPro with Adagio, Finn and Lance vs Ringo, Glaive and Ardan - (from 0:35:50) Play Video
Another video from Vispro - (from 1:39:06 & 2:12:55) Play Video

I hope you find that guide usefull. You can also share any expirences with Adagio - fell free to comment. In the future i will add roam section to that guide. GL in your next match with Adagio!