Electricity needed to maintain power for camera
You've got a few misconceptions about the GoPro's battery and how power is supplied to the GoPro itself. I'll try to clear those up and then explain how you can do what you're trying to do.

In your third bullet point, you claim the camera is rated at 1030mAh and 3.8V. Those numbers are actually referring to the nominal voltage and charge capacity of the GoPro's battery, not the camera itself. The unit mAh (pronounced milli-Amp-hour), or just Ah, is used to describe how much capacity a battery has. It is not a measurement of instantaneous current (Amps). The different between Amps and Amp-hours trips up a lot of beginners and it's worth spending some time understanding the difference.

So why does this matter? Well, the GoPro is a sophisticated piece of consumer electronics. It's designed to maintain and charge its own battery. You don't need to worry about charging the battery yourself. It already contains all of the circuitry internally to do that for you. All you need to do is provide it with the voltage it expects to get from the outside world. After some quick Googling, I see that the GoPro expects to be plugged into a USB cable. That's a good thing! USB voltage is a very common 5VDC. You just need to provide a steady 5V source and the GoPro will do the rest.

The only unknown that's left is how much current the GoPro will draw while operating. The worst case scenario is a fully depleted battery and recording video at a high resolution. These two conditions combined will result in the largest current draw from the 5V source. It looks like someone did an interesting experiment to discover the power consumption of various GoPros in various different modes. Using the information in that website, it looks like the high-end of current draw is about 1A. Add a little extra current in case the battery is charging and let's round up to 1.5A*.

We have the voltage and approximate current, which is enough to calculate the power consumption of the GoPro while recording video and possibly charging the battery from a 5V source:
$$P=VI=(5V)(1.5A)=7.5W$$
Now you know how big the solar panel needs to be. But there's still one more thing you need to consider. The solar panel won't output exactly 5V. In fact, the voltage will vary with the sun intensity over the course of the day. But the GoPro needs a steady 5V or you risk damaging your expensive camera.

The solution is to pick a solar panel that generates a voltage higher than 5V. For example, 12V solar panels are very common and easy to find. You then connect the output of the solar panel into a high-efficiency voltage regulator that converts the voltage down to 5V. Now you've got a steady, regulated 5V to plug into your GoPro. Even if the 12V solar panel varies over the course of the day, the regulator will adjust to keep its output at 5V.

Obviously once the sun starts to set or if it's too cloudy, all bets are off on the regulator's ability to maintain the necessary voltage. You'll need to come up with some solution to prevent an under-voltage condition. But that's beyond the scope of this answer. 

*I'm applying an educated guess to come up with this number. There are a large number of unknown variables about the internal circuity of a GoPro that could render this number completely wrong.