One of the new Pokemon added in Generation IV was Lickilicky, the evolved form of the Generation I Pokemon Lickitung. Lickilicky's evolution is slightly different than other previous generation Pokemon evolutions that were added to the Sinnoh region. Why do we say this? Well, it's about learning a specific move.
To evolve a Lickitung into a Lickilicky in Pokemon Brilliant Diamond and Shining Pearl, you will first need to get the regular Pokemon Lickitung. So, you will need to teach Lickitung a very specific move. The move is a Rock called Rollout.
This move is simple enough to learn if your Lickitung has been bred, as the regular Pokemon will actually learn it at Lv. 6. However, if your Lickitung doesn't know this move and Lickitung has passed that level, don't worry.
You will need to go to the Move Tutor to teach Rollout to Lickitung. The Move Tutor is located in Pastoria City. However, you'll need to beat Maylene, Sinnoh's third Gym Leader, to get to the Move Teacher. And make sure you have a heart scale. A heart scale is required to relearn or teach a Pokemon a move via the Move Tutor.
Related: Where to Find a Heart Scale in Pokémon Brilliant Diamond and Brilliant Pearl?
Once you've taught Lickitung the Rollout move, you're not done yet. You will have to level up Lickitung once again, while the Normal-type Pokemon knows Rollout. This can be done by fighting other Pokemon and earning EXP, or just with a rare candy.