"What is it about the word king that makes even the most reasonable people behave like idiots?" Ivar asked King Harald before the first battle of Kattegat. He was trying to convince Harald to help him defeat Lagertha so he could become king. His words were like a prediction of the king he would make, though, as he became one of the worst kings in Vikings.
When Bjorn became king, he was the exact opposite of Ivar, taking Kattegat back to what Gunhild called the golden days. The two were sons of Ragnar, but while power turned one into a heartless tyrant, it taught the other to be humble and wise. Such is how kings behaved in Vikings, some totally unfit and useless while others understood their people's needs and brought a positive change with their reign.
Kjetill Flatnose
Kjetill named himself the King of Greenland, officially making himself the worst king in the entire show. Wherever he went, Kjetill caused trouble, and his first action as king was to sentence people to death by starvation. He wouldn't have been made anyone's king in Norway because he was a tyrant that didn't know how to live with people, let alone lead them.
When he followed Floki to Iceland, he had the ambition of starting life in a new place and fulfilling his ambition of becoming king. He then committed a massacre in Iceland after gaining Floki's trust. He also gained Ubbe's trust on the trip to Greenland before nearly getting everyone killed because he wouldn't share a beached whale.
Ivar
Ivar used Aslaug's murder as his excuse for hating Lagertha and taking Kattegat from her, but he had other plans. When he became king, he started behaving like the same idiot he had told king Harald about. He turned into a tyrant destroying everything good that Ragnar had created in Kattegat.
After declaring himself a god, he killed the seer and then proceeded to close off Kattegat from the rest of the world, telling the people that foreigners were a threat to them. The worst came when he started burning people alive for dissenting against his oppressive rule. He was a good fighter with great battle strategies thanks to Ragna's training, but he made a terrible king. Kattegat didn't really achieve anything good from the whole time that Ivar ruled it.
King Aelle
King Ecbert was shrewd, and Aelle suffered at his hands multiple times, but most of Aelle's suffering was caused by his own hand. When the Vikings came to Northumbria during Ragnar's second raid, Aelle never took his time to study and understand them before attacking. He could even have gone to ask for Ecbert's help, but he chose to attack them on his own, causing the embarrassing defeat.
He then foolishly helped Ecbert take Mercia for himself before being humiliated in public by his own daughter. When Ecbert finally brought a captured Ragnar to him, he never imagined that it could be a trap. Ragnar's sons finished whatever was left of his kingdom when they came to avenge their father.
King Horik
It was Horik's idea to leave Jarl Borg behind on the trip to Wessex, then when Borg took Ragnar's lands, Horik left Ragnar to come back alone to fight for his rightful place. The least Horik could do, was respect the Earl who had made him famous by taking him to England.
Horik instead chose to eliminate Ragnar using his best friend, Floki. Horik had absolutely no idea just how powerful Ragnar was. His influence as king was never felt while Ragnar lived, and that is how he died. He went down as the most insignificant king of Kattegat.
Aethelwulf
If Aethelwulf had welcomed Ubbe and Hvitserk and agreed to give some land to the Vikings, he wouldn't have spent his entire reign as king losing battles to Ivar. He never had the chance to make a single kingly decision. He was a failed king from the moment his father gave him the crown of Wessex. Aethelwulf had spent his entire life serving as Ecbert's puppet without learning any actual leadership skills.
He failed to listen to Haehmund when he warned him about engaging Ivar and ended up losing all his battles against Ivar. Ecbert's son also refused to negotiate when Ubbe and Hvitserk came to offer terms of peace away from Ivar's madness, but he wasn't as smart as Ecbert. It is not clear what killed him, but whatever it was, it saved Wessex from certain doom.
King Olaf
"Come and see the wreck of humanity," Olaf said as he burned Bjorn's men alive when they came to rescue King Harald. There was little to no explanation to most of the things King Olaf did, but he always worked for the greater good, including helping Bjorn topple Ivar, but not before unnecessarily torturing Hvitserk.
He was a wise old king who knew what Norway needed, but his madness overshadowed most of his goodwill. Compared to the likes of Horik and Ivar, Olaf was a good king. He fell victim to Harald's tricks and Oleg's cruelty in the end, but he accomplished the one thing no other king did in Vikings; Uniting all of Norway under one crown.
King Harald
Harald Finehair surviving until the very last episodes of Vikings was one of the miracles in the show. He was a weak king that only tagged along on Ragnar's raid of Frankia to gain some fame. Ragnar and the rest of his family never trusted him, but he was a lesser evil compared to the alternatives the Lothbroks had. He knew how to play his cards which helped him survive even after Ivar betrayed him.
He was a weak leader, though, and he had no vision of what Norway was supposed to look like. When he had it all, he still lost as Bjorn accomplished the mission of defeating the Rus while he languished in Oleg's cell. Ivar then soiled Harald's legacy by convincing him to go on a doomed raid of England. He left the leadership of Norway more broken than he found it.
King Alfred
Alfred wasn't as good a fighter as Aethelwulf nor cunning like Ecbert, but he was a wise king who always admitted his shortcomings and breached them with knowledge and training. He was the perfect king for Wessex when it needed to live with Vikings while fighting other invaders.
It was a huge burden that even the best fighters like Aethelwulf and Aethelred couldn't handle. He strengthened Wessex's defenses using Ubbe's training to defeat future raids by Vikings while still offering a window for peace by granting the piece of land in East Anglia. His visionary leadership and positive thinking allowed him to defeat Ivar accomplishing his grandfather's mission.
King Ecbert
King Ecbert was no match for Ragnar on the battlefield, but he managed to prevent the sacking of Wessex by Ragnar in his lifetime. He knew when to attack and when to negotiate and always got what he wanted in the end. He was shrewd and cruel and ended up hurting his son and the rest of his neighbors, which is why he doesn't get the best King's spot.
His tactics worked for the overall good, though, and he managed to keep Wessex safe for long enough for his son to have something to inherit. He was smarter than Aelle, always knowing how to avoid unnecessary bloodshed. His sins caught up with him in the end, but even in his last moments, he managed to save Wessex.
Ragnar
Ragnar Lothbrok was just a farmer in Kattegat that ended up becoming the most famous Viking in the world. He had no leadership experience when he started raiding. All he wanted was a good piece of land for his people to settle and farm. He didn't accomplish his missions, although his sons did, and the credit goes back to him.
He failed in some ways, especially by underestimating the threat posed by his brother and the inability to unite his family before he died. Ragnar's accomplishments overshadow all his mistakes. He set a precedent for massive achievements that his people could never have imagined.
Bjorn
When Bjorn Ironside died, Gunhild declared that the age of heroes had ended and that the golden days of the Vikings were over. He was the last great Viking king and also the best of them all. He was reasonable enough to take Kattegat from Ivar without attacking his own people.
He went on to lead his people against the Rus, and although he lost the first battle, he managed to rally all of Norway to fight off the greatest threat they had ever faced with the last bits of his breath. He was alongside Ubbe in Wessex when they got the land for Viking settlers, so he gets credit for that too.