You get a kitten by finishing the Gertrude's cat[runescape.wiki] quest and after that you can buy more kittens to raise from her after finishing the quest.

1. Drop the kitten on the ground from your inventory to have it follow you around.
2. Let if follow you for 2.5 to 3 hours - ish
3. You will get some dialogue messages every once in a while telling you what the kitten needs. The message will not appear in chat, again it will be a dialogue message.

• If it wants attention, right click on it and interact with it and stroke it
  
• If it's hungry, feed it any type of cooked fish, raw fish (tuna, sardine, etc), or bucket of milk. One unit of food is enough. You get milk using an empty bucket on a cow. Milk counts as food. It can grow on fish only. It seems that any 1 unit of food resets the hunger to 0.
  
• I noticed that if you stroke it and feed it every 10-15 minutes or so, the dialogues don't pop up.

4. You can check the kitten's growth progress and hunger by opening the summon/familiar/follower window.

5. Feed it to keep the hunger low. If it reaches 100% it will run away and you will lose it.
6. If you're a dog person, the kitten will take longer to grow into a cat.
7. At 100% growth, CAT!

1. I'm trying to do the Varrock medium task Faster, Pusseycat! Kill! Kill! [runescape.wiki] where I need a cat training medal from Gertrude. Can I use the kitten to catch 100 mice?
Don't. You'll need a grown cat. Kittens almost never catch mice. You can right click to interact Chase vermin when there's mice nearby (Lumbridge Castle), but it will almost always attempt and fail. I've never managed to. Wait till it grows. After you get an actual grown cat, it has 50% chance to catch the mouse, so it would take approximately 200 attempts.
2. If I kept my kitten dropped, tagging along, for one hour and I pick it up and logout or wait a while, will the growth progress reset and have to start over?
No, you can safely pick it up and drop it later, the progress timer will resume. If you plan to literally afk, I'd advise picking it up just in case.
3. If I'm skilling, can I accidently click while the dialogue is popping up and miss it?
Yes, doing an action closes the dialogue message and you might miss it. Keep in mind food is not an issue beacause you can monitor that, but attention is. When in doubt, just stroke it!.
4. My kitten is constantly meow-ing, am I doing something wrong?
No, he's just being a little stinker.
5. If I keep the hunger percentage low, will it grow faster?
Not that I've noticed

• The kitten will teleport to you when you stray too far. It will teleport where you teleport like a summon. Don't worry about that.
  
• You can sell the cat for 100-200 death runes to West Ardougne citizens (because they have rat problems?) Cats grow into overgrown cats after a while. They can't catch mice anymore, but they can still be sold in West Ardongue.
  
• You can't trade cats on the GE and you can only hold one kitten at a time in your inventory/bank. You can have multiple cats.
  
• You can use a ball of wool to play with it but I'm not sure if that counts for its interaction need.
  
• I don't know how many interaction dialogue messages you are allowed to miss before the kitten runs away because of lack of attention. :(
  
• Let me know if I missed anything critical or correct me if I got anything wrong.