I swear by Apple (no relation to the Apple computers) brand products. For all my leather products, I rub and buff them with the Apple conditioner and then spray them with the Apple guard protector. For the aniline leather you want to use a poly spray rather than a silicone spray, especially if your case is any color other than black. The Apple products don't darken or change the color of the cases. You can also use the conditioner to clean. Once protected, you shouldn't get any deep stains so wouldn't have to deep clean. That's the key to aniline leather - you have to protect it. If you don't, once dirt or grime soaks in, it's extremely hard to pull it out of the leather without also pulling out the dye.
What's nice about Apple leather product, is that it doesn't leave any sticky residue. Just feels soft like after you have moisturized your hands.
FYI I have no affiliation with Apple leather - just found out about them through great reviews from a purse blog.