Mimicking an adapter with a dc power supply
Voltage is the first thing you need to consider. You must find a regulated voltage source which matches what your laptop requires. The next step is figuring out if your voltage source has a current source which can supply enough amperes to power your laptop (in this case, it should be able to provide at least 2.42A). So what you're looking for is a power supply which outputs 19V, and can deliver at least 2.42A. If it can deliver more, that's fine. Your laptop will only pull as much as it needs.

If you do hack something together, beware that you will be working with mains power source. Also, make doubly sure that you don't get the polarity wrong.

I don't know if 20V is alright or not, you have to check what range your laptop can tolerate. I will claim, without justification, that 20V should work fine.

You can use batteries, of course. It depends a lot on what kind of batteries. If you plan on wiring up a bunch of AAs and such in series/parallel to get what you want, you may, but they probably won't last long.

My suggestion would be to buy a power adapter for your laptop. They're very common and easy to find, and will save you a lot of trouble/concern for safety.