First, how many Elder Scrolls games have you played? I am assuming only Skyrim, as you list that game as his home continuum, rather then Elder Scrolls. That is not something that needs to be fixed, necessarily, but I felt I should point it out. Then again, Za was only used as a prefix in Arena and Daggerfall, so perhaps you played more then I thought.
Where is Za’kiir from? Was he born in Elsweyr, or, well, elsewhere? If Elsweyr, which region? The vast deserts of Anequina, or the jungles of Pelletine? This is important, as it may help with personality and culture. The Khajiit from Anequina are desert nomads, fighting in tribes amongst each other. Think Arabic influences. Meanwhile, Pelletine is the center of trade, nobility (such as Khajiit have nobility, save the Mane) and production. If from Pelletine, Za'kiir may be a merchant, or a moon sugar producer (the main export of Elswyr), or a guardsman.
However, what if Za'kiir hails from another land? Cyrodiil, or Morrowind, or even Skyrim? Then, ask yourself, why was he from there? Was he born in one of those provinces? Did he travel there for trade? Became a sell-sword? If from Morrowind, was he a slave who recently found freedom, due to the collapse of the Dunmer nation? Or, was he from Elswyr, and was banished to one of the other lands?
You mentioned he has a pension for hoarding shiny objects. Where did this come from? Both meta-contextually, and in-character. I have a slight suspicion this arose from the stereotype about Khajiit being a race of thieves. Don't worry, the stereotype is mostly true. However, I want to be sure you understand why they have this reputation. First, and foremost, is because of Khajiiti ideas of ownership. Let me paint you a short scenario.
Lets say a Khajiit man is hired at a local good store. The shop owner tells him to move some inventory upstares. The Khajiit does so, and continues doing so for a while. Eventually, he gets hungry. Upstairs, he sees a fish on a barrel. Being hungry, he takes the fish, and eats it. When the shopkeeper finds out, he fires the Khajiit and has him arrested for theft. Khajiit is confused as the guards drag him off to prison. He stole nothing! He only ate a fish!
To you and me, it is clear that the fish belonged to the shopkeeper. However, to the Khajiit, it was just a fish out in the open. Had the fish been labeled that it belonged to the shopkeeper, then the Khajiit would not have taken it. Well, at least, not without compensation. As it was, it was just a dead fish out in the open. Who owns every fish in the world? It belonged to nobody, and so now belonged to Khajiit.
However, more pertinent to you character, is the second ideal of Khajiit. That of cleverness. The cat people of Elswyr pride themselves on cleverness, and one way of showing that cleverness is to take things that belong to others. What matters to the Khajiit is intention. Did they steal it for the thrill of taking what belongs to others? For showing how light their fingers, and quick their paws? For the glory of breaking through impenetrable defenses? Then, all is forgiven, and Khajiit is praised for their cleverness. However, if it is out of selfishness, or greed, then they are among the worst of Khajiiti society. They could even be banished, should it be heinous enough.
Basically, what I need is back story. Even if we never directly see it, or have it only slightly alluded too, or even never mentioned at all, it is still a good idea to have the back story in mind. I love the Elder Scrolls seres, and would be happy to help you through the lore, if you need it. Short of that, I will need to see him in action before I can juge further.
