Is there a specific way to brush so that whitening tooth paste is more effective?

No specific way: The goal of brushing your teeth is to remove bacteria from your teeth and gums, to prevent cavity and gum disease. Tooth paste helps in the above action however the teeth whitening claims may not happen.

Yes & no: Tooth whitening toothpaste often have a dual effect. There may be a chemical that has an effect but the paste may also have extra abrasives that whiten. You need to be careful not to use a toothpaste that is too abrasive.

Circular motion: Brush in a circular motion slowly so the bristles get the entire tooth.

Sustain contact: For a whitening toothpaste to be effective, the toothpaste needs to be contact with tooth to allow time for the agent to whiten your teeth. Professional whitening using a tray to allow the gel to be in contact with your teeth for a period of time to allow the whitening gel to whiten your teeth. The longer the whitening gel contacts your teeth, the better the whitening results.