• health (has it ever injured itself? is it fit? does it show signs of lameness)
• potential (what levels do i want to ride and is the horse i am looking at physically able to reach these levels)
• quality (what have the horse's parents achieved? does the horse have a strong quality jump? does he have nice gates?)
• handling (is this horse good to handle? does it have good ground manners? is it easy to load? does it bite or kick?)
• experience horse and rider (how experienced is this horse? how old is it? am i experienced enough? if the horse is difficult, can i handle it's behaviour?)
• riders feeling (do i feel well riding this horse? do i feel safe or does it scare me?)
• inner voice (does this horse impact me positively? what does does my inner voice say?
• aesthetics (do i like how this horse looks? do i like his movements? what do his eyes say?)
I would definately recomment trying the horse before purchasing it. By best friend bought a horse online from the netherlands that had been successful in the grand prixs and had won a lot. She didn't try him first. He ended up hard mouthed, panicky during courses and dangerous to handle. She was very dissappointed. So i would't recommend doing it that way.
I also wouldn't buy a young inexperienced horse if you can't handle it. But if you have a lot of experience and want a "project horse" that can be perfect too.
Temperament and what you are looking for. Don't try to trick yourself
Focus on what your ultimate goal for the horse is. Then go through the aspects that discipline requires. look for that in the horses. Also, be careful when looking online! Many of the ads are scams! And don't be afraid to make a list of questions to call and ask before you go look at the horse. I think the best horses are the ones that are not finished yet. This enables you to take it in any direction!
First of all the soundness, then the brain, experience, movement. Since I got several really good ones, I would also consider about the bloodline.