First of all, he's adorable!
Playing--this takes some time, and encouragement! You have to help them learn to play with toys. I get on the floor with a basket of foot toys and play with my birds--chase them with the toy, dangle it over them. It won't take long until he is chasing the toy back, flipping over on his back so you can play with him and actually bringing toys to you so that you can play with him. Caiques are very interactive, and don't just want to necessarily play alone. That's one reason I like having two--they definitely prefer playing with each other, or with me!
Purring--everything
@rockybird said.
Hopping--mine really didn't start hopping until about 8-10 months old. It takes some coordination! Again, showing them the behavior will encourage it. Pick up a toy and hop it across the table, or the floor. If you and your caique have a high level of trust, gently pick him up and hop him across the floor. The more you can physically interact with your caique the better.
Mimicing--Start with a simple two-tone whistle. This is easiest when you use it as an alternative to their contact call. If you are outside the room, and your bird calls to you "scree..scree..scree..", Whistle back. When he copies your whistle, enter the room and say "GOOD BIRD!" Repitition and consistency is the key. Only work on one sound at a time, until they start practicing it themselves. I started saying "Whatch doin" everytime I entered the room and engaging them. Now, when I leave they ask "whatcha doing". Also when they're looking for me. Mimic the pitch and cadence of a sound they make naturally. Caiques will never be excellent talkers like an Amazon, but YOU will understand what they're saying, and they will use it in the correct context!