If Iluvatar created the world with the Fire that Melkor had sought, why didn't Melkor use that part of the Fire to become even more powerful?
In the Ainulindalë (Silmarillion), the author tells us that Melkor had no access to the Fire (Imperishable Flame):
To Melkor among the Ainur had been given the greatest gifts of power and knowledge, and he had a share in all the gifts of his brethren. He had gone often alone into the void places seeking the Imperishable Flame; for desire grew hot within him to bring into Being things of his own, and it seemed to him that Ilúvatar took no thought for the Void, and he was impatient of its emptiness. Yet he found not the Fire, for it is with Ilúvatar.
The implication here is that the Fire is an expression of the creative force of Ilúvatar, and not a separate existing talisman. Although when Ilúvatar creates Ea "the Secret Fire was sent to burn at the heart of the World," this does not change the nature of the Fire itself as an expression of Ilúvatar's creative force.
This isn't about degrees of power. Melkor was already the most powerful being, not just in Middle-Earth, not just in Arda, but of all the Ainur, as Iluvatar himself confirms:
Then Iluvatar spoke, and he said: "Mighty are the Ainur, and mightiest among them is Melkor"
The mightiest of those Ainur who came into the World was in his beginning Melkor
(I'll skip over "in his beginning" for now but it will come up again later)
And he shut the doors of his house in the face of the mightiest of all the dwellers in Ea
This shouldn't leave Melkor's status in any doubt, this is one powerful dude we're talking about.
But power wasn't what Melkor sought; as Rob's answer quotes:
desire grew hot within him to bring into Being things of his own
Even assuming that the Secret Fire had the ability to confer more power (and that's a very big assumption, as the sole ability it has that's attested in Tolkien's works is that of creation of other beings) that's not his objective; Melkor - as Morgoth - was far more interested in domination, and - when he couldn't have it - destruction.
The "Note on Motives in the Silmarillion" essay in HoME 10 has much bearing on this, and is recommended reading, but in case the OP doesn't have a copy of it handy, I'm going to quote a few passages:
Thus, as 'Morgoth', when Melkor was confronted by the existence of other inhabitants of Arda, with other wills and intelligences, he was enraged by the mere fact of their existence, and his only notion of dealing with them was by physical force, or the fear of it. His sole ultimate object was their destruction.
This was sheer nihilism, and negation its one ultimate object: Morgoth would no doubt, if he had been victorious, have ultimately destroyed even his own 'creatures', such as the Orcs, when they had served his sole purpose in using them: the destruction of Elves and Men.
even left alone he could only have gone raging on till all was levelled again into a formless chaos
Morgoth had no 'plan': unless destruction and reduction to nil of a world in which he had only a share can be called a 'plan'.
All of this still leaves open the point: if Melkor just wanted to destroy, surely more power would have better enabled him to do so, so the OP's question remains valid. The answer to that is quite simple; he had already tried and failed. Again, from the same source:
though he had 'disseminated' his power (his evil and possessive and rebellious will) far and wide into the matter of Arda, he had lost direct control of this
This is where the "in his beginning" element I passed over above comes in; in order to exert his dominance over the matter of Arda, it was necessary for Melkor to disseminate his power into Arda itself:
Melkor 'incarnated' himself (as Morgoth) permanently. He did this so as to control the hroa, the 'flesh' or physical matter, of Arda...in this way Morgoth lost (or exchanged, or transmuted) the greater part of his original 'angelic' powers, of mind and spirit, while gaining a terrible grip upon the physical world...The whole of 'Middle-earth' was Morgoth's Ring.
So in order to attempt to gain control over Arda, Melkor in fact reduced himself to a quite impotent state in which he no longer had domination over that which he put his spirit into; even if the Secret Fire had been put right in front of him, he lacked the power to actually do anything with it, and that's even assuming that the Secret Fire had any capabilities other than that of creation (and if - as seems likely - it doesn't, the ability to create is of little or no interest to a being hell-bent on destruction).