I think the Ner'zhul-Arthas combination is something that will get discussed in the book. From the little preview from awhile back, it seemed like Arthas was still trying to be a little good and Nerzhul evil. But the whole Horde going for Ner'zhul and Alliance going for Arthas is something that could happen.

I hope this happens when the lore expands in later content patches...

At the moment, the Lich King is still very much "Arthas", although wowwiki begs to differ:

When asked, "What's the truth about the new Lich King?", Chris Metzen replied: "Arthas and Ner'zhul have become a perfect fusion of one being - Arthas' personality and body with Ner'zhul's wisdom, experience, power and EVIL."

After peering into the Lich King's heart, Tirion Fordring remarks that "there's nothing left to redeem" and "there is no Arthas Menethil anymore -- there is only the Lich King".

I haven't quested all the way through Icecrown. I should probably do that before I start making assumptions about the Lich King eh?

the horde doesnt, and probably wont focus on defeating ner'zhul because there are very few people in Azeroth (especially horde-side) that know that ner'zhul is involved at all.
the few orcs that may still remember ner'zhul likely have no idea what happened to him after he jumped through one of his portals. pre-arthas almost everyone just knew him as 'the lich king'

just thought id clear that up

There are a few Forsaken quests that mention that they know about Ner'zhul, so I'd assume it's something that most of the Horde leadership is aware of.

From what i can gather from the various questlines and the wrathgate cinematic, the alliance and horde only think of Arthas, the fallen human prince, the Knights of the Ebon Blade however truely know, since they once served him, and the ebon messenger (i think the name was) in early phased icecrown, tells you that the ligh king "is 2 of the most ruthless and cunning entities to have existed", so no, from what i see, neither the horde or the alliance realize that ner'zul is the big guy, and just want to smash arthas' head in.