Elder Scrolls Arena and its health system

This is something that really makes Elder Scrolls Arena awkward for me: the health system. From the starting dungeon I got killed like six times before breaking free. And the rest of the game is the same: you get killed easily even by rats.

The first time I've played this game I took a male nord warrior character and started to increase his Endurance and Strength so I could take more hits and carry more stuff, also I got plate armor by level 5. But I've died hundreds of times before reaching level 5 which pissed me off a lot, also in order to finish the game I had to cheat my health points and increase them by almost 2000 so I could complete the main quest.

How does the health system work on Arena? I hate to be killed by two strikes from goblins or rats. I don't know if I've been distributing wrong my skill points, or the game it's just that difficult.

The health system is not overly complicated, and yes the game is challenging.

As seen in the wiki on the health system in TES Arena:

"Health represents the total amount of damage the Eternal Champion can take before dying in The Elder Scrolls: Arena. Maximum health can only be increased when the player levels up, and the amount received is based on endurance, but is also random."

Ways to heal according to the same page are:

"Health can be restored by using potions, healing and stealing spells or magic items, as well as resting or being healed in a temple. Health can also be protected by using shielding spells."

Gathering that there is nothing crazy about the health system, it could just be that the game is challenging and you have to better manage your health.

That being said, here is a General Hints Guide on early levels of TES Arena. Some key points related to your question: (Bold emphasis mine as they relate most to your problems early)

• Save now, save often! There are no event-driven saves like auto-saves or quick-saves in the game. Generally speaking, you should save the game whenever you kill something, find new treasure, rest and recover, and before you open a door.
• Every time you explore a little further, note where there are resting areas.
• Attacks will often be ambushes from behind. Always look around you and never assume the creature you just killed is alone. Keep the sound up since monsters usually make a standard sound particular to their type either before attacking or when they come into range (a kind of leitmotif).
• Always find a raised area to rest on until you have fully recovered. You will never have random encounters if you rest on any raised surface, no matter how little it is raised, which means you can rest as long as you need to without being disturbed. If you rest on the floor, you may be subjected to attack from a random monster. It's advisable not to proceed forward until you are completely recovered and then save the game once you have.
• Game play is fairly unbalanced. With low-level characters, the Warrior class that can wear plate armor has a significant advantage. Characters that can wear plate armor, any of which can be found with enchanted properties, potentially have seven additional magical enhancements over non-plate-wearing characters. For those just starting out and getting the feel of the game, you might consider a Warrior that can wear plate armor for survivability.
• As you progress and level up, the monsters encountered will be more dangerous. However, if following the main quest line, the game play can become unbalanced as certain types of monsters will always be found in certain dungeons and low-level characters can quickly find themselves overwhelmed, getting killed in one or two hits, or unable to cause any damage to certain monsters. While it is possible to go through the entire main quest below the 20th level or so, virtually every battle towards the end can be a frustrating life or death struggle.

Judging from your experience and the guide (and having to 'cheat'), I think you're progressing farther than your character level and fighting things too strong for you, hence dying so much. So try out some of these tips and see how that works out!