If you want the desperation of their descent into madness to draw sympathy and understanding, then I'd look at the pillars that maintain human life. Food, water, and a safe and stable place to sleep. If you threaten only one of those, it's going to create stress for long as that pillar is threatened. We can't survive without food and water, and our mind never gets a break without knowing we are safe, and that the safeness is stable. Being homeless, for one, is horrible. Anyone can become homeless, and once you're homeless you know that some nights or mornings or certain areas at certain hours throughout the day, are safe, and you can sleep, but there's no sense of stability. There's no guarantee of anything, not even safety while you're starving. Even the fear of becoming homeless will put extreme amounts of stress on people.
People are great with adversity and adaptation. We can learn to deal with all kinds of systematic abuse and never fight back, but if you threaten one of those three things, I think that's when we really start to crack as a whole, and certainly on the individual level.
For those with loved ones, you can add in a fourth pillar, which consists of family or friends. Some characters may not take huge risks for themselves, but they would to take care of their little brother, or their daughter, or protect an ailing grandparent, or their best friend.
Then, looking at those and the stress it puts on people, to be able to promise any of that to victors in a challenge can definitely drive competitors to the ring to do what they need to, in order to live comfortably. The more dire the circumstances the less questionable their decision to enter a battle royale is.
There's also the cultural element. While Hunger Games' battle royale was enforced by law, there one to two districts (I don't remember which ones) that adopted a warrior-attitude. They were excited, and honored to join. It was part of their culture, and I believe kids were trained at an early age. They were conditioned, and at that point, didn't know any better as they'd never been shown anything else. That's an option, where some of the characters had an entirely different cultural upbringing and don't think of the violence or atrocities people are forced to go through in a battle royale as bad.
Another thing that could add to the credibility is if people straight up don't even know exactly how dangerous the battle royale is. There could be an iron-clad contract you have to sign--it's up to you to read. Little is heard about the battle royale because it's pretty secret. All you know is a lot of people don't come out or disappear. Maybe you tell yourself it's some conspiracy--it has to be, so that you can talk yourself into doing something you're certain will help you in the long run, if only you win. So you sign the dotted line and...quickly regret that decision 
.
As for the general "What makes a death tournament story good?" I'd say it's high-stakes. You can't get much more high-stakes than death, but it's important to care about the characters first. I like how Hunger Games did it. I did not like how Battle Royale, the manga, did it.
(Spoilers for BR) There was no time spent with the vast majority of the characters who died. They'd be introduced while in the disaster, then it would show a chapter on their tragic backstory, and then they'd die. It felt more like someone briefly introducing a puppy then killing over and over, trying really hard to tug on my heartstrings, but going way too fast for me to notice. While that definitely works for some readers, I think spending more time with characters and developing them would have worked for more readers. But, of course, that's not coming from any heavily-researched place or anything, it's my take ^^
In addition to that, exploring the nuances of how stress and mental unwellness affects and controls us is something I find is rarely touched upon in these high-stake situations. It often becomes a question of right or wrong, but we're not built to be under extreme stress and I feel like the toll that takes on people and how it blinds us against our will is severely underestimated. Some may say there's always a choice. But how can there be a choice if your brain is so unwell at this point that you literally couldn't have made any other choice except for the one you've been trained to make. 
It's not a necessity, but it's always a nice thing to touch on, to be a bit more judgement-free of characters. Hunger Games (novel) was one of the rare instances for a battle royale situation where I felt none of the competitors were demonized. The movie, however, definitely demonized some.
