Your pet's tier is determined by the sum of your pet's stats: health, strength, defense and speed, so you increase their tier by increasing their stats :) You can do this through the Training Center, winning battles to earn EXP to spend on stats, or using stat boosters

Check out the first 2 columns in this chart to see what stat totals you need to reach each tier. (Keep in mind intel and lvl do not count towards tiering up)