There is no surefire way to increase your Dark Souls attunement slots, but there are a few things you can do to improve your chances.
First, try to play as many different characters as possible. This will help you learn which ones are the best for your playstyle, and which ones offer the most bonuses.
Second, try to find a good covenant or alliance. These groUPS of players can offer help and support, as well as provide you with extra resources and bonuses.
Finally, make sure to always be leveling up your character. This will increase your attunement slots, and overall stats.