Get a Kong Buddy without purchasing it from Funky Kong's shop
This requires two controllers. Simply add a second player from the World Menu before you start a level. Press Start, then select Add/Drop. Press A on the second controller. Then choose your control options. Finally, select between Diddy Kong, Dixie Kong, or Cranky Kong. From here, you can play with a second player, or remove the controller from the Add/Drop menu. You will still have your Kong Buddy.
Hard difficulty
Complete all three stages on Secret Seclusion Island.
King K. Rool reference
There is a rock formation in the background of level 3-1 "Grassland Groove" that resembles King K. Rool.
Playing with the 3DS
If you left Donkey Kong alone without having him do anything in Donkey Kong Country Returns, he would pull out a DS and play it. In Donkey Kong Country Returns, his idle animation is similar, except now he'll play with a 3DS instead.
Secret Seclusion Island
Complete the special K level on all islands.
Special K level
Collect all the "KONG" letters in all levels on an island to unlock the special K level.
Various Unlockables
By collecting the KONG letters in every level on an island, a new stage will unlock.
Unlockable
How to unlock
Hard Mode
Complete the 3 stages on the 7th island.
Island 7: Secret Seclusion
Complete the special K level on each island.
Zelda Reference
Like you could in Donkey Kong Country Returns, you can find Funky Kong and choose the Crash Guard item to see him say "It's dangerous to go alone, take this!" as a reference to the classic NES game, The Legend of Zelda.