Hi Jessup, and welcome. What an excellent question, I expect this one will keep us happy for quite a while!
Bantie keeper has put it in a nutshell, and I'm sure you'd notice immediately if any of your hens was off colour or unhappy. I think hens are not so different from us, though they may express it differently - they are happy if they are healthy, appropriately fed, suitably housed and sheltered, with enough room to explore and do what comes naturally, and in an environment which is largely free from either social stress or the fear of predators. If they have these conditions, they will show the signs of health Bantiekeeper has summarised. But I'd just like to add a bit about the effects of stress in hens on their happiness and health.
If something seems wrong, if a hen looks unwell or is isolated and bullied by the flock, as keepers we need to know enough to sort it out, (or preferably, to have prevented it happening in the first place, by good husbandry.) We all learn from our mistakes, but a lot of problems can easily be avoided by researching and putting into practice a setup which provides suitable shelter and space, free from fear of predators, appropriate food, clean water, a regular routine, and being accepted socially into the flock without being bullied. I wouldn't be happy if someone I lived with kept pecking my head and making it bleed! Also, freedom from stress can be affected if the hens aren't chased or picked up in the wrong way (perhaps by excited children) or barked at by dogs, even if the dog is the other side of the wire and poses no actual danger. Keepers do need to be able to handle their hens, so they can be regularly examined for parasites or injuries, and know what action to take when needed, but hens soon become tame enough so this can be done without stressing them, if the keeper always moves and speaks quietly and consistently in the henrun. Stress has an enormous effect upon the health of hens, and thus on their happiness. As in humans, stress and illness are often linked. It's generally accepted that hens are more likely to become ill shortly after being moved to an new environment, especially if they have been through what, to them, are traumatic experiences like being taken to market, spending a day in a cramped cage in crowded, noisy surroundings, probably with inadequate food and water, followed by a journey in a vehicle, before being released into a totally new environment, possibly with a lot of strangers. This is likely to depress their immune systems so that they are more likely to succumb to whatever they've picked up by way of viruses etc en route. So getting them from a breeder, bringing them home quietly straight away, and giving them time to recover in isolation before introducing them to the rest of the established flock is a really good idea, for everybody's sake.
i think you just know a happy, relaxed, alert hen when you see one, - the important thing is, to be able to notice when one isn't happy, diagnose the problem, and do your best to make things better again.