Those kits certainly let you rush a batch of wine to the bottle, but in my opinion it doesn’t make for the best wine. I do those kits a couple times per year just to play around with it, but a lot of the “extra” ingredients are not necessary. You’ll certainly want to keep a minimum amount of sulfite in the wine, but I’m sure the kit came with a packet of clarifying agent, some sorbate, and maybe some other chemicals. Do a little bit of research about what those ingredients do, and you’ll be able to decide for yourself if you want to use them.
My suggestion, once primary fermentation is complete, rack it to a secondary fermenter and let it sit. It’ll drop sediment for a long time, which might be sped up by something like super-kleer KC, but there will be lots of tannins and other harsh compounds that might take some time. When you rack it, make sure you top up to within an inch or so of the stopper so you don’t have more oxygen than necessary in the headspace. A similar wine can be used, or in a pinch, distilled water. Wait until you can shine a flashlight through it without any cloudiness, and rack it again to sit longer or to bottle. If you bottle from a container with some sediment, it’ll be really tough to keep it out of the bottle. Every other racking, or right before bottling, you’ll want to add more sulfite if you choose to use it.