You can check it three ways.
Method 1 — The Pokétch
Use the Pokétch app that looks like the one below to check to see if your Pokémon has max happiness. If you tap and hold your Pokémon and it gives two large hearts, then its happiness is maxed.
Method 2 — The Footprint Man
On Route 213 you will find a house with a man inside. Show this man one of your Pokémon, and he will look at its footprint and guage its happiness levle. If it's maxed out he will give your Pokémon a Ribbon.
Method 3 — The Hearthome City Fan Club
Talking to one of the Fan Club members will have her tell you the happiness leve lof your lead Pokémon.
Source: Experience (for the Hearthome thing)