Interestingly enough in other 'fantasy' logic, to 'kill another person' is one of the forbidden wishes/limitations, that a genie cannot actually grant. The limitations of genies are somewhat different in KQ lore (see Guidebook to the Land of the Green Isles).
A genie does have some limitations, however: it cannot cure ills, change the weather or bring back the dead.
It says of the Prince who became the master of Mali Mellin;
"His enemies lay slaughtered on the fields without provocation."
Other, than that the 'motives' and 'reasonings' behind it are actually mentioned or implied in the game at various points. It's also discussed in the KQ6 Hintbook, IIRC.
One place it may be mentioned in the game is those cutscenes with Alhazred issuing orders to the genie (after you trade for the magic map). There are two or three versions of that scene (depending on if you meet Alhazred first, or if the genie informs Alhazred of your coming to the island).
In those scenes we find that Alhazred doesn't directly order Shamir to kill him. He tells him to spy on him, and he tells Shamir to tell the other leaders of the islands to look out for him and stop him by any means necessary. You learn later, that Shamir had described Alexander as an 'assassin', 'saboteur', or 'spy' among other things to convince the island's leaders to stop him.
In a later cutscene, after Alex fakes his own death. Alhazred doesn't believe it at first, and thinks Alexander is extremely devious and may have just tricked him. But is glad that he's gone, if its true. He calls it 'convenient'. But there is still no indication, that he directly ordered the Genie to "kill Alexander".