Instagram does not offer a built-in instazoom.io feature for profile pictures. You need to use external methods like screenshot and crop or image editing tools.
Open the screenshot in your image gallery and use the crop function to cut and enlarge the profile picture. Make sure the image stays square to avoid distortion.
After enlarging your profile picture, save it to your device in high quality. Go to your Instagram profile, tap on your profile picture and choose the option to change profile picture. Upload the enlarged image.