Feanor was never fond of Fingolfin. The seeds of discontent were sown long before Melkor was unchained.
Feanor's family history is extremely unusual.
Then Finwë was grieved, for the Noldor were in the youth of their days, and he desired to bring forth many children into the bliss of Aman; and he said: ‘Surely there is healing in Aman? Here all weariness can find rest.’ But when Míriel languished still, Finwë sought the counsel of Manwë, and Manwë delivered her to the care of Irmo in Lórien. At their parting (for a little while as he thought) Finwë was sad, for it seemed an unhappy chance that the mother should depart and miss the beginning at least of the childhood days of her son.
She went then to the gardens of Lórien and lay down to sleep; but though she seemed to sleep, her spirit indeed departed from her body, and passed in silence to the halls of Mandos.
Miriel died in Aman. Feanor grew up without a mother.
All his love he gave thereafter to his son; and Fëanor grew swiftly, as if a secret fire were kindled within him. He was tall, and fair of face, and masterful, his eyes piercingly bright and his hair raven-dark; in the pursuit of all his purposes eager and steadfast. Few ever changed his courses by counsel, none by force.
We might suspect then that he was a little spoiled by this attention.
But what never, ever happened among the Eldar, before, and very seldom after, was this: Finwe remarried:
Now it came to pass that Finwë took as his second wife Indis the Fair. She was a Vanya, close kin of Ingwë the High King, golden-haired and tall, and in all ways unlike Míriel. Finwë loved her greatly, and was glad again. But the shadow of Míriel did not depart from the house of Finwë, nor from his heart; and of all whom he loved Fëanor had ever the chief share of his thought
The wedding of his father was not pleasing to Fëanor; and he had no great love for Indis, nor for Fingolfin and Finarfin, her sons.
(Of Feanor and the Unchaining of Melkor)
In other words, Feanor hated all his half brothers from the very beginning. To an elf, this was positively unnatural. Elves did not remarry. Indeed, afterwards, many among the Eldar blamed Finwe for what happened:
In those unhappy things which later came to pass, and in which Fëanor was the leader, many saw the effect of this breach within the house of Finwë, judging that if Finwë had endured his loss and been content with the fathering of his mighty son, the courses of Fëanor would have been otherwise, and great evil might have been prevented; for the sorrow and the strife in the house of Finwë is graven in the memory of the Noldorin Elves. But the children of Indis were great and glorious, and their children also; and if they had not lived the history of the Eldar would have been diminished.
(Of Feanor and the Unchaining of Melkor)
We can refer to HoME for a few more details. For example, Finwe had to get a special ruling from the Valar to do this:
Since death and the sundering of spirit and body was one of the griefs of Arda Marred, it came inevitably to pass that death at times came between two that were wedded. Then the Eldar were in doubt, since this was an evil unnatural. Permanent marriage was in accordance with elvish nature, and they never had need of any law to teach this or to enforce it; but if a ‘permanent’ marriage was in fact broken, as when one of the partners was slain, then they did not know what should be done or thought.
In this matter they turned to Manwë for counsel, and, as is recorded in the case of Finwë, Lord of the Noldor, Manwë delivered his ruling through the mouth of Námo Mandos, the Judge.
‘Marriage of the Eldar,’ he said, ‘is by and for the Living, and for the duration of life. Since the Elves are by nature permanent in life within Arda, so also is their unmarred marriage. But if their life is interrupted or ended, then their marriage must be likewise. Now marriage is chiefly of the body, but it is nonetheless not of the body only but of the spirit and body together, for it begins and endures in the will of the fëa. Therefore when one of the partners of a marriage dies the marriage is not yet ended, but is in abeyance. For those that were joined are now sundered; but their union remains still a union of will. [...] It is therefore true to say that, though achieved by and in the body, marriage proceeds from the fëa and resides ultimately in its will. For which reason it cannot be ended, as has been declared, while that will remains.’
So Manwe made a judgement: a marriage cannot be broken unless the dead Elf never returns to life. Which means that when Finwe remarried, that Miriel was dead-dead.
While there is hope or purpose of return it is not ended, and the Living cannot therefore marry again. If the Living is permitted to marry again, then by doom Mandos will not permit the Dead to return. For, as has been declared, one reborn is the same person as before death and returns to take up and continue his or her former life. But if the former spouse were re-married, this would not be possible, and great grief and doubt would afflict all three parties.
(Laws and Customs among the Eldar, HoME II)
This was done with some degree of Miriel's consent. But as we saw above, even if she changed her mind, she couldn't return to life.
Mandos had spoken his doom as has been recorded,3 Manwë called Finwë to him, and said: ‘Thou hast heard the doom that has been declared. If Miriel, thy wife, will not return and releases thee, your union4 is dissolved, and thou hast leave to take another wife.’
(Laws and Customs among the Eldar, HoME II)
So Feanor has a pretty understandable reason to hate Fingolfin and Finarfin. Jealousy, in that Finwe used to be focused entirely on him, which was fair, considering that losing his mother after childbirth was something considered incredibly grievous by the Elves - they preferred to not have children rather than risk raising them alone. And on top of that, the very existence of Fingolfin was a reminder that Miriel, Feanor's mother was dead forever and could never come back. Which is why he moved out ASAP:
As soon as he might (and he was wellnigh fullgrown ere Nolofinwë was born) he left his father's house and lived apart from them, giving all his heart and thought to the pursuit of lore and the practice of crafts.
(L&CoE, HoME)
Morgoth simply exploited the feelings Feanor already had towards his half-brothers by planting the rumor among the Elves that Fingolfin planned to usurp him and Finwe. This was already something Feanor believed them capable of, because he was blinded by his feelings.