The brothers are
the vowels a, e, i, o, u of which in English, there are five.
Without us you'll strive...
It is impossible to write meaningful English text without using vowels.
Eldest always believed in us...
U, being the furthest along in the standard English alphabet, believes in us, or rather, in Us (where the -s is a suffix to the root u)(this is wordplay)
Next one believes in nothing
(So remaining four said him goodbye)...
O appears similar to the cardinal 0, zero, representing nothing. O itself is also contained on the word "nothing".
This combined with the hint for U/us leads me to think that believes is read as a pun on pronunciation, it be lives in ....
Finally, the remaining four vowels but not O itself, are used in " adieu*, French for goodbye (per the questioner, smriti's comment).
Youngest three together, are not so cruel or unkind,
But sometimes they can be insane...
The remaining three vowels are A, E, I. All are needed for, and thus sometimes in, the word "insane" but are not sufficient to make "so cruel or unkind" as this would need O and U.
When together, we will be airy,
Though we are likely to fall apart...
Vowels are often linked together as diphthongs - as an example, in the word "airy". However, words spelled this way may have the vowels pronounced separately. The vowels themselves are also spread quite separately throughout the standard English alphabet.
This might also be a subtle pun. A homophone of "airy" is "eerie" - and when all the vowels are placed together in order, they make a very eerie ghostlike word, "aaeeiiooouuuu!"
But controlling our misbehaviour...
The vowels must follow a number of rules, some of which can be labyrinthine. The case of "I before E, except after C and in some arbitrary other weird words" is an easy example.
We will try to be your favourite.
The word "favourite" contains all five of the vowels!
Additionally, the vowels are very overrepresented in use compared to their proportion of the standard English alphabet.