When you want to say that you have to have something or that you need to have something that POINTS TO A CAUSE (ex: you need something to complete a task), you would use: -なければいけません/ないといけない
Rather than saying, "I need to have something," in Japanese, we say, "If I don't have something, then it (the situation) can't be," or, "If I don't have something, then I can't go on [to complete this task]."
And remember contractions for ease of pronunciation: (Formal) なければいけません --> ないといけません //// (Informal) なければいけない --> ないといけない
So, let's say you want to bake a cake, but you don't have eggs. You would say, "たまごがないといけない。” or "たまごがないとできない" or "たまごがないと作れない" (you get the point, right?)
However, in your situation, since you want a cat JUST BECAUSE, saying, "ねこがないといけない" would sound a bit unnatural because it sounds like you're saying, "If I don't have a cat, then I can't do ____." So, instead, it would sound more natural to say, "ねこがほしい" (simply "I want a cat"). Or, if you want to be more dramatic, you can say, "ねこがないと生きていかれない" ("If I don't have a cat, I can't go on living.")
Also... There is no kanji required for this since "なければいけません/ないといけない" is purely Japanese and has no Chinese influence.
Hope this helped!