Even though the iPhone doesn't have a built-in heart rate sensor, you can still find a third-party app for that!
Apple has put a much larger emphasis on health and fitness in iOS 8 than in any other version of iOS — yet the iPhone doesn't offer a heart rate monitor. That being said, there are third-party apps in the App Store that can pick up the slack, and don't even require an external sensor of any kind!
How to check your heart rate on iPhone without an external monitor
In order to monitor and track your heart rate on your iPhone, I suggest picking up one of the following App Store apps:
I personally use Instant Heart Rate by Azumio because I also use Fitness Buddy (another one of their products) to build custom workouts. I haven't tested Runtastic's version as thoroughly, but it's similarly accurate to Instant Heart Rate. If anyone has more experience with it, however, be sure to drop your thoughts in the comments!
Heart rate monitor apps for iPhone work by having you place your finger over the camera lens of your iPhone. It uses the camera to monitor your pulse, then provides you with a heart rate estimate. I've being using Instant Heart Rate for quite a while now, testing it with both with gym equipment and by manually calculating my heart rate. It's typically within 2-3 beats so I'd consider it fairly accurate.
Instant Heart Rate also lets you do a sit/stand test that can test the health of your heart. While I don't take too much stock in these tests, as they're definitely not true medical assessments, they do provide some interesting data points. And if you ever think there's a problem, the more data you have to present to your physician, the better.
Instant Heart Rate can also integrate directly into the Health app to add your heart rate readings. So if you aren't using a fitness tracker that has a built-in heart rate sensor, a heart rate monitor app can make a great stand-in.
Give one of them a try and let me know in the comments how they work for you!