Death from Below:
This is going to be your main attack, and will generally be used to initiate a fight unless you're lucky enough to be able to get close to the enemy without it. I level this one up first and frequently because it serves as your entrance to the fights and it applies a blight on all targets hit. This blight will enhance the effects of all your other attacks. Death from Below is also your main crowd control attack, followed by Devour Souls.
Shroud of Darkness:
This attack is of limited use at lower levels and without a blight applied, but once leveled up, and when used properly, it can have amazing offensive or defensive potential.
Defensive
Under normal laning circumstances, it can be used to prevent hit and run attacks by weaker enemies, saving your health for larger offensives. An enemy will skirt the fighting creeps, jump towards you and try to launch a follow up attack, but you can silence them before they get it off. They will be forced to retreat because you and your lane partner will have removed their speed/surprise advantage.
Offensive
As an offensive ability, this is paired to great effect with Death from Below, which applies a blight on hit. This blight then causes a silence AND fear when you use Shroud of Darkness. At higher levels, this gives you 2 seconds to launch an attack or vacate the area, using it as a hit and run or a pre-cursor to your ultimate. This also works well when you know someone has an escape; you can generally get them to pop it prematurely by using these two moves together.
Devour Souls
This attack is a good one for crowd control but also has uses as a finishing move if your ultimate doesn't quite kill a god. For a little extra damage, make sure that a blight is applied. Also keep in mind that when a blight is applied, targets hit with this heal you. While it isn't enough to make much of a difference late game, this can be used early game to save you a few health potions: Death from Below into creeps and use this to heal up (plus deal additional damage to them).
Pillar of Agony
This is one of the best ultimates in the game. First, while beads will allow people to escape its embrace, the damage is dealt over time, not at the the end (like Ymir's), so .5 seconds is all it takes to damage someone at least a little bit. Also, I have found that most people don't think to buy beads when facing Hades. I don't know why, but rarely do I find someone that uses beads to escape. Having said that, an escape is very effective at avoiding Hades' ultimate. My strategy is to usually wait until an enemy has used their escape to
start an attack and then go through the kill combo.
Attack order:
Death from Below
Shroud of Darkness
Devour Souls
Basic
Pillar of Agony
Kill Order:
Death from Below (unless you are close enough in which case use your basic attack and save this)
Pillar of Agony
Devour Souls
Shroud of Darkness
Then Since this seems like a really good guide in depth except there's no icons or colors.