Why Hanuman did not become/considered a king even though his father was a king? Who ran the kingdom after Kesri (hanuman's dad) if hanuman did not take his place?
Who is Hanuman's father?
Hanuman's father name is Kesari who was a valiant Vanara. He is not the biological father of Hanuman. Hanuman is born in the womb of that vanara's wife Anjana by the grace of wind God.
In Sundara Kanda, sarga 35, Hanuman explains the characteristics of Rama and Lakshmana and then gives a brief explanation of his birth.
कौरजो नाम वैदेहि गिरीणाम् उत्तमो गिरिः |
ततो गच्छति गो कर्णम् पर्वतम् केसरी हरिः ||
"O Seetha! There is a mountain called Malyavan, which is the best among mountains. From there, a monkeys called Kesari went to another mountain called Gokarna."
स च देव ऋषिभिः दृष्टः पिता मम महाकपिः |
तीर्थे नदी पतेः पुण्ये शम्ब सादनम् उद्धरत् ||
"As prayed by gods and seers, that great monkey called Kesari, my father killed a demon named Shambasadana at a sacred shrine near the sea-shore."
तस्य अहम् हरिणः क्षेत्रे जातो वातेन मैथिलि |
हनूमान् इति विख्यातो लोके स्वेन एव कर्मणा ||
"O Seetha! I am born in the womb of the wife of that Kesari the monkey, by the grace of the wind-god. By my activity alone, I am renowned as Hanuma in this world."
Was he a king?
He is not a king but a lieutenant in the vanara army. When we observe the verses in which Hanuma mentions Sugriva, he uses the word king. But he didn't use the word king when we told about his father. For example,
वयम् तु हरि राजम् तम् सुग्रीवम् सत्य सम्गरम् |
परिचर्यामहे राज्यात् पूर्वजेन अवरोपितम् || VR 5-35-26
"We for one are serving that Sugreeva, the Lord of monkeys, who was true to his promise, but was dethroned from the kingdom from his elder brother."
In other instance, Vanaras who were happy after knowing that Hanuman had been successful in his work in knowing the whereabouts of Seetha goes to Madhuvana, a garden owned by monkey lord Sugreeva. They assault the guards and the keeper of the garden Dadhimukha. Dadhimukha is also maternal uncle of Sugreeva. He was unable to stop the vanaras from drinking the honey and damaging the garden. He decides to complain this to Sugreeva in Kishkindha.
In the complaint, that Dadhimukha says as follows:
न एव ऋक्ष रजसा राजन् न त्वया न अपि वालिना |
वनम् निसृष्ट पूर्वम् हि भक्षितम् तत् तु वानरैः || VR 5-63-4
"O king! Madhuvana was indeed not set free previously at any time by Riksharajas, your father, or by you, or by even Vali. That Madhuvana has been consumed away by the monkeys."
By the above shloka, it is clear that it was Sugreeva's father Rikshirajas was on the throne before Vali. Vali and Sugreeva acquired the throne by ancestry.
So, Kesari was not the Lord of monkeys but a Valiant Vanara who had killed Rakshasas. Not being a king doesn't defame his valiance or the greatness of Hanuman. The story which you have said in the question looks like a made up tale by television show writers to exaggerate Hanuman's birth and early life before meeting Sri Rama.
All translations from Valmikiramayan.net