Step 1: Purchase Pokémon Bank.
It's five greenbacks for a year, and renewable. Available from the eShop.
Step 2: Download Poké Transporter.
Downloadable from the Pokémon Bank home screen.
Step 3: Get two DS systems. Insert Platinum in one and Black/White/Black 2/White 2 in the other.
Having access to 2 DS's and one of BW or BW2 is critical to transferring Pokémon to ORAS. If you do not have these then unfortunately you cannot transfer.
Step 4: Follow the steps in the link.
I answered this section a while back and it's easier to link it than write it out again. Once finished everything in the link, proceed to the next step here.
Step 5: Put your BW/BW2 game in the DS with PokeBank.
Step 6: Open PokeBank, select "transfer" and select your game.
This will allow you to transfer entire boxes of Pokémon from your game into the PokeBank.
Step 7: Select Alpha Sapphire in PokeBank and deposit your Pokémon.
Voila! The process is complete.
Assuming you followed all of these steps correctly, you should now have your Platinum Pokémon in your Alpha Sapphire game. Enjoy!