Space in the PC for that/those Pokémon to be stored.
Another Pokémon in party.
Two Pokémon that are not banned from the Battle Tower (can be cloned).
This is the PC in the Battle Tower.
Go to the PC in the Battle Tower.
Deposit the Pokémon to be cloned in Lanette's PC.
Log Off of the computer and save the game.
Withdraw the Pokémon stored in the PC from step 2.
This girl is who you need to be talking to.
Talk to the receptionist for the Link Multi Battle Rooms (she should be at the counter closest to the PC).
When she asks if the player wants to make a challenge, select Yes, and choose the two Pokémon eligible for entry.
Select Yes when she asks you if it is OK to save the game.
After the 2-3 seconds where no messages are shown and the game (not the receptionist) asks if the player wants to save the game, reset the game (do not start saving).
How it works
The game is saved by the player after the Pokémon are deposited. During the time when nothing seems to happen, the game saves the Pokémon in the player's party along with other data such as planted berries and Day Care Pokémon, but not those in the PC. Thus the game now has the Pokémon in both areas.
Make sure to follow the steps exactly and do not withdraw or deposit any Pokémon besides the ones to be cloned, or else Pokémon could be deleted.
It is possible to clone many items at one time, simply by replacing step 2 with "Give items you wish to clone to any Pokémon in the PC", and replacing step 4 with "Put the items back in the bag".
By starting this glitch without any Pokémon in the Day Care, the player can duplicate up to eight Pokémon at a cost. The player must start with one Pokemon in their Party that they are okay with deleting. By first withdrawing two Pokémon from the PC and putting them in the Day Care, then withdrawing five more Pokémon from the PC, next swapping the Pokémon to be deleted in the Party with the final Pokémon to be cloned from the PC, and finally returning to the Battle Tower and proceeding from step 5 onward.