If little has been recorded of Horus Lupercal's early years, then Alpharius is truly enigmatic by comparison. There are multiple accounts of his rediscovery, all of which may or may not be true in part or whole. Added to this is the existence of Omegon, and potentially others who could command the Alpha Legion and are its true master(s). All that said, I like the line in the background text of Alpharius' entry the states "when the time comes to cast off the cloak of misdirection, Alpharius is as awe-inspiring a being as any of the Primarchs of the Legions Astartes".
Lets be clear from the outset. If you are wanting to play a primarch who exists just to beat other players or other primarchs up, then Alpharius is not who you are after.
Alpharius' strengths are two-fold: firstly as an army wide booster (or force multiplier), and secondly for directly messing around with the enemy like no one else can.
His major boost to his sons comes in the form of Preferred Enemy: Everything. This is truly spectacular. I cannot underscore enough what this has the potential to do for all of his sons. Given the first turn, and close enough range (which is very likely due to the Alpha Legion rule of Mutable Tactics), this is as close to an alpha-strike (no pun intended) as I can see being pulled off by any 30k army. And that's before a "win at all costs" player thinks about automatically exploding tanks out of nowhere. As a minor boost, out flankers can also add an extra bit of movement on the turn of arrival.
He messes around with the enemy in a number of ways. Seizing the initiative half the time can, in certain situations, be a game winning situation for the Legion. This could be added with other initiative seizing boosts for great effect. Further to his insidious mastermind rule, he can bring in his own units from reserve if the enemy successfully brings one in on their turn and Alpharius has a matching unit. The wording of this rule is that it must be of the exact same entry type. Hence legion terminators from the Alpha Legion can come in to play if the enemy is trying to deep strike their own legion terminators. It doesn't matter if they're armed differently, or if they have different numbers -- just the unit entry type has to match. This is important as it encourages the Alpha Legion player to think about taking certain units that might ordinarily go in to reserve in their army if they're playing Alpharius. As well as legion terminators, this would also include scouts coming in from outflanking, as well as drop pods (etc.).
In addition to this, he can "hide" inside on of his own units at the outset of the game, only to reveal himself later on. This replaces a regular line trooper and can rally a squad at an opportune moment. However, given that his preferred enemy: everything cannot be active unless he is revealed, there is a temptation to have him revealed from the first turn.
Although he is clearly no slouch in close combat, he is also not in the same league as Angron, Vulkan or Horus. To be clear: he's probably going to die to all three of these primarchs in a fair fight, and to many of the others besides. But this is not the reason to play Alpharius. You should be playing him for his army force multiplier effects and messing with the enemies' mind.
Alpharius has an interesting combination of equipment. For armour, he is 2+/4++ and cannot be affected by poison or flesh bane (perhaps he just silently disliked Mortarion and the Death Guard?). Unlike a number of his brothers, he actually carries a ranged weapon in the form of a plasma blaster (master crafted, naturally). As well as venom spheres, he also has the pale spear which from the fluff appears like it might be some type of necron artefact -- its interesting for its combination of armour bane and instant death coupled with AP1 at Alpharius' own strength and initiative. He can clearly decimate enemy units with this, but his brothers will still wipe the floor with him most days of the week. He therefore shouldn't be fighting fair!
He also has some minor upgrades like cameloline and the combination of a nuncio vox and Cognis Signum. This suggests he should be placed with a shooting orientated squad -- perhaps even Lernaean terminators?
Alparius is an excellent addition to the Alpha Legion and the brotherhood of primarchs. He is unique in his combination of rules of messing with the enemy and being a force multiplier. But he's not powerful enough to go toe to toe with Horus et al. Since when did Alpharius fight fairly though? His sons should have already ripped apart an enemy force and opposing primarch before Alpharius gets close to finish the job if I'm honest. And Alpharius' rules beg to be utilised as part of an Alpha Strike decapitation effort. And he's actually moderately cheap compared to some of his brothers as well. Take him and/or Armillus Dynat to be complete (but remember that Dynat's hammer strike warlord trait is lost since Alpharius has to be the warlord of an Alpha Legion detachment).