How to get more stamina by running

There's no stamina bar, the only exception is when you do the Triathlon race.
One way to find out when you are running low on stamina before starting losing health during "normal" activities it's by looking at the health bar, it starts flashing before starting draining your health.

According to the official GTA 5 Strategy Guide, here is a comprehensive list:

• Stamina: this skill is increased by 1% for every 17 meters (18 yards) ran, every minute swam, every minute cycled. Participating in triathlons is a great way to increase this ability.

• Shooting: this is increased quicker by landing hits on enemies, scoring headshots and successfully completing Shooting Range challenges. This skill is increased by 3% for every Gold challenge completed, 2% for every Silver and 1% for every Bronze.

• Strength: this is increased 1% for every 20 punches. Playing sports (golf, tennis, darts) also increases strength.

• Stealth: this skill is increased by 1% for every 45 meters (49 yards) walked in Stealth Mode and by 1.5% for every two stealth kills.

• Flying: flying for 10 minutes increases this skill by 1%.
  Completing the following training at the Flying School increases this skill by 3% for Gold, 2% for Silver and 1% for Bronze: Training take off, Runway landing, Inverted flight, Knife flight, Flat hatting, Loop the loop, Sky diving, Drop zone.
  Completing the following training at the Flying School increases this skill by 5% for Gold, 3% for Silver and 2% for Bronze: Touch down, Helicopter course.
  Completing the Helicopter speed run training increases the stats by 6%/4%/3%.

• Driving: this skill is increased by 1% for every 10 seconds of wheelie action and an additional 1% for every second spent in the air in a vehicle or four-wheel landing. Performing Stunt Jumps with four-wheel landings gives the greatest boost to this skill.

• Lung Capacity: this stats is increased by 1% for every minute spent underwater.