Using super capacitor alternative to battery as a backup power for laptops
As RedGrittyBrick pointed out, the biggest problem is energy per volume. Batteries of pretty much any sort store far more energy than ultracaps for a fixed volume. Caps have definite advantages, such as lifetime (a good ultracap should be functional long after a battery won't hold a charge any more) and power density (peak current out of an ultracap module can be thousands of amps!). But for energy storage, batteries win, hands down.

That doesn't mean it's not an interesting idea! It just means you need to know what to expect. Assuming you want to replace your internal battery entirely, your operational lifetime will be cut, probably down to maybe 5% of what you had, if you're lucky. Now, for some applications, the difference between four hours and ten minutes is a killer. For others, all you really need is enough time to travel between power outlets. So even the reduced lifetime isn't really a deal-breaker.

Also, if your goal is to replace the internal battery, your new module will need some brains to talk to the BIOS properly. Otherwise, the PC may not think it has a battery at all. To avoid that complexity, you might be better off bypassing the internal module connections and plugging into the laptop's DC input. The PC would think it was running on external power, so you'd have no software indication of how much power you had left. Again, depending on the application, that may not be such a big deal.

Now, if you plan to leave the internal battery in place, and just charge that off the external ultracap, I'd suggest that's not worth doing. The amount of energy stored in your standard internal battery is, oh, let's say something like 160 kJ. (Rough estimate, 90W power supply, 30 minute charge time.) An ultracap to hold that much energy is much bigger than a car battery, and outweighs your laptop by probably a factor of twenty.

In summary, this is probably only worth doing if you replace the battery entirely, and don't care about reduced runtime. And even then, probably only as a fun project, and not as anything practical.