If he really wants to be a king of Asgard he could easily prove himself to his father that he is more powerful than Thor
instead of killing Odin.
It shows that he has no humanity for his father.
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Loki killed* Odin for numerous reasons.
Loki is consumed by jealousy and anger, largely stemming from a lack of approval by Odin and jealousy of Thor.
Thor is a hero. Loki is a disgraced criminal who has been defeated twice.
Having been defeated by Thor twice already, killing Odin is the only other way to claim the throne.
*as we learned later, Loki did not kill Odin, he kidnapped him and hid/stranded him on Earth.
Despite what we see at the end of Thor: The Dark World,
Odin is not dead. Loki lacks the raw power and the chutzpah to take out Odin, All-Father, and Ruler of Asgard.
More than likely this is another deception by Loki to rid himself of the burden of his step-brother, Thor while cementing his rule on the throne.
Loki has about half of the attributes necessary for being a good king. Unfortunately for Odin, Thor only has half as well. I suspect Odin would prefer Thor to rule, not necessarily purely from Thor being of his bloodline, but because when all is said and done, Thor is necessary for Ragnarok not to be a complete rout of the Asgardians. Left to Loki's rule, Thor would not survive until then. Without Thor, Ragnarok is lost, no Asgardians survive and a new age of gods (and men) does not happen.
Odin knows this because he sacrificed his eye at the Well of Mimir and has done everything in his power to ensure Thor's will be leading the Asgardians at this final day.
What most people don't know about Odin is he embodies the powers and spiritual essence of his other brothers, Vili and Ve, and that power is known to Asgardians as the Odin-power or Odin-Force.
Not only does this mean Odin is far more powerful than the averages Asgardian, his abilities have few equals in the Nine Realms. Loki, while one of the most powerful sorcerers in Asgard, is not an equal of Odin in any way.
Upon Odin's passing (as it has happened on more than one occasion in the canon comic series) this power will pass to the strongest descendent of Odin. That would be Thor.
Since that has not happened yet, and Loki being aware of magic, is probably cognizant of this, the last thing he would want is for the Odin-power to find its way to Thor until he can claim it for himself.
Knowing Loki, he has imprisoned Odin somewhere away from his tools (Gungnir, Spear of Heaven) and somehow blocked Odin's access to his own prodigious magical abilities. It will likely fall to Thor, as it has so often in the comics, to rescue his father from Loki's innovative and dangerous prison.
What Loki often fails to remember is, any place strong enough to imprison Odin, is certainly strong enough to keep Loki locked away until a week before Ragnarok...