Pretty much any decent laptop will run DJ software fine. If you are doing basic stuff a I5 8GB Ram & 256GB SSD would work. I use virtual DJ for everything, karaoke, DJing, VJing & production stuff. But since you don’t have a karaoke library I would continue with Karafun for now but think about upgrading to the Pro plan and be legal. I have not used Karafun as I have my own library, its look to have everything. I Believe you do need an Internet connection but I think it might also work off line if you download the songs.
With karaoke I’d say the big cost would be buying your own karaoke library or continue to just rent until you see how far you want to go with it, can you make any money with it.
I started as a DJ and got asked to do karaoke because I had laserdisc players and large video screens. Over the course of 25+ years are turned into a big part of what my company does, ironically I can’t sing, never really sung unless I’m in the car/van trying to stay awake. My kids described it as a dying cat or worst.