Explanations: PT Str and 3 bracers are all applied to you and your illusions, thus giving you 532Hp and 28dmg to you and Illusions also. Plus, illusion gain agility and you have bigger mana pool.
Manta is for additional ownage and confusing enemy. Also, all +stats are applied to illusions.
Heart is no brainer.
Bottle is if you are ganking all the time.
Rejected items: Vlads gives you 0/nothing/zero. Its a complete waste of gold for small lifesteal which only works on you (yes, illusions don't gain anything, its fake animation). Battle fury is also a mistake, since all illuisons gain NOTHING. Only you gain. Complete waste of gold. Same goes with HotD. Life steal only works on you, so all you've got is small lifesteal, nothing else.
^That should probably do it.
Cp6uja explained it all nicely, there really isn't much to add Have fun! =)
Pers will generally delay your other items but if you feel there's a need to get one you can go ahead.
SnY if youre owning, bkb againist disables, Maelstrom if you need more steady dps. heart is good luxury.
anyway, core is just bracers and treads. thats all that really matters.
on topic: hes right about the core. afterwards its a good idea to rush a HOT since ur images benefit the most from it. armlet also not too bad, remember to activate it before split. sny is ok. bkb not that great on nessaj imo, but situationally its needed. the quoted post would actually be a decent advice... if it wouldnt be for the maelstrom o.O
edit: wow that fancy post with pictures just caught my eye.. ye bottle is good, close to core on CK. forgot that :S i personally dislike diffu on him, the purge is nice but not needed, the manaburn isnt all that great on his images since theyre more for towerrazing than anything else, the agi.. doesnt hurt but.. yea. str would be better :P manta isnt terrible, but id rather buff the awesome images he has than adding gimped ones. heart before it imo. didnt try new one on him yet though.
off topic: reduced CD on his ult would be nice and simple buff. agreed?
It gives you AND your images a huge damage boost, and for the first 13 seconds you actually have MORE health than you otherwise would, and your illusions permanently gain +25 strength providing you activate it before you split.