No matter how good a cellphone camera is, you won't get good quality videos until you increase the amount of ambient light around the subject. To be able to get better videos out of any decent camera, you need to include better lighting.
You can buy numerous lighting setups for your video recordings, but the best source of good lighting is the Sun. I would suggest you to record a few sample videos on the subject outside, in the Sun and let us know if the video quality is upto your requirement.
Even under the Sun, you would need to make sure you are following the basic photography techniques, like avoiding backlight, using angles to improve lighting on the subject and avoiding harsh afternoon sunlight by using filters etc. Also, please do not use after-market camera apps from the Play Store and use the default camera app on the cellphone to get the best out of it. You can always post process the video (if you need to) using a video editor.
All the above assuming, you've kept the phone's video recording parameters to the maximum possible settings. Samsung Galaxy S3 comes with one of the best produced cameras out there and is capable of recording full HD 1080p videos at 30fps. I suggest making full use of its 8 MP camera by applying the suggestions above.