Either your monitor is out of calibration or your iPhone screen is out of calibration. Ideally you should calibrate both. I don't know if you can calibrate an iPhone 5s screen.
The iPhone OS is color managed but there's nothing you can do to 'calibrate' and profile that device. Shouldn’t be necessary. Your own display is a different story.
First, open a color reference image with known, good quality RGB numbers that should preview well. Here's a couple of examples but they must be viewed in a color managed application (like Photoshop, Lightroom, etc):
Examine the colors, perhaps recalibrate your desktop display if necessary. Convert them to sRGB and upload on the iphone. Do not expect a perfect match! This may help too:
sRGB urban legend & myths Part 2
In this 17 minute video, I'll discuss some more sRGB misinformation and cover:
When to use sRGB and what to expect on the web and mobile devices
How sRGB doesn't insure a visual match without color management, how to check
The downsides of an all sRGB workflow
sRGB's color gamut vs. "professional" output devices
The future of sRGB and wide gamut display technology
Photo print labs that demand sRGB for output
High resolution: http://digitaldog.net/files/sRGBMythsPart2.mp4
Low resolution on YouTube: https://www.youtube.com/watch?v=WyvVUL1gWVs
Thank you. It was the monitor calibration!