Martin says that only one of Septim blood could wear it, but Mankar was quite clearly wearing it, both in Paradise and in the shrine on Tamriel when you first see him

There is no official answer but the common theories are:

#1. Mankar could wear it because he has Ayleid heritage and the Amulet of Kings, or Chim El-Adabal, was originally an Ayleid artifact.

#2. Mankar is dragonborn (often said that only those with "dragon-blood" can wear the amulet and this can sometimes be translated into "dragonborn"). The theory about him being dragonborn comes from a passage in his Commentaries.

#3. Mankar's family has familial ties with the Septims and so he is therefore distantly related to them.

Also to note on #3: it is likely that a large portion of the nobility in Cyrodiil has Septim blood so the list of potential wearers could be quite long. Secondly, those who wore the amulet in the earlier Empires/dynasties were not Septims. Finally, I can't remember where I read that the Camorans were somehow related to the Septims so I could be wrong about that. And that brings us to...

#4. It's all Imperial propaganda and that you don't need to be "of Septim blood" to wear the amulet. Of course the CoC can't wear it so there is obviously some requirements to wear it. I personally like to think that to wear it you need to truly believe that you are "a king", "an emperor" or a "ruler" and that this is how Mankar was able to wear it.

Also note that this question has been asked before and you can read four of those threads http://www.gamesas.com/topic/1237986-what-is-dragon-blood-and-why-is-it-required-to-wear-the-amulet-of-kings/, http://www.gamesas.com/topic/1254353-how-did-mankar-david-cameron-manage-to-wear-the-amulet-of-kings/ and http://www.gamesas.com/topic/864831-the-amulet-of-kings/ and don't forget over http://www.gamesas.com/topic/868884-the-amulet-of-kings/ as well.

Daggerfall had something similar: only those of royal/noble blood could wield Numidium, excluding the player but, strangely, permitting an Orcish chieftain. This might be some sort of obtuse divine favoritism.

Of course, its perfectly possible for Mankar to have a drop of Septim blood: the Camoran dynasty has been a respected bloodline since the First Era, and it would be a natural move for the Septims to reinforce their hold on Valenwood by marrying off some second-rank cousin of theirs.

The amulet is sentience enough to choose who wears it. Uriel Septim obviously can somehow commune with it... or the souls within it.

Perhaps Mankar did his weird cray cray Oblivion magic enough to convince the amulet to let him wear it.

I like this. It is my new canon.