Kearny's opinion of McClellan was not sourer than his opinion of everyone else. I have
a copy of Kearny's letters to his wife and it's an amazing thing to read. What is continually reiterated is his opinion that he is the best commander in the US, and there is a grand conspiracy to keep him down. At one point he even accuses his uncle of conspiring to prevent him have a brevet in 1851 for an action of his company in Oregon which he was in New York during.
The Kearny-McClellan relationship soured more than normal after Johnston abandoned the Centreville-Manassas position. Kearny's brigade was the brigade posted on the observation line that day. On detecting the movement he didn't tell anyone and simply advanced with his brigade towards Fairfax Court House. The next day he pushed through to Centreville, and claimed (wrongly) to have been the first troops there. He latter claimed to his wife to have chased away the whole rebel army with one regiment.
Kearny's lack of communication had delayed the Federal pursuit by a half a day . However, McClellan's cavalry were in Centreville and beyond to Manassas hours before Kearny reached Centreville. Three regiments had passed through - the 2nd New York Cavalry, 3rd and 8th Pennsylvania Cavalry. The 2nd NY Cav took the captured stragglers back to Washington whilst Averell pushed the 3rd and 8th Penn onto Manassas and "kept a sword in the enemies back".
Kearny's feud really began over this - he said he was first into Centreville and claimed the glory. McClellan noted the cavalry was there before him.
Things rapidly got worse. A few days later McClellan attempted to promote Kearny to a division. Kearny objected to the division offered, and Lincoln intervened as to whether Richardson could have the division instead. McClellan simply noted Kearny's refusal and appointed Richardson instead. This was more grist for Kearny's fantasy that the whole army was engaged in a conspiracy to keep him down.
The soured relationship continued to get worse, especially after Williamsburg when McClellan accidently gave Hancock his "the Superb" sobriquet.
For his part, McClellan liked and defended Kearny. In his memoirs McClellan admitted he should have made Kearny a cavalry commander instead - the infantry didn't suit him. Kearny hated McClellan, because in Kearny's eyes McClellan was preventing him being promoted Major-General.