You could use either word. 'The character of modern youth' means the following sums up everything about them - their whole character. 'A characteristic of modern youth' means the following is one aspect of their character but there are other aspects too. Note you'd have to say 'a characteristic' because people have many characteristics but only one character. So it really depends on what you want to say.

I'd go with 'character.' 'Characteristics' tend to be more narrow, descriptive, or even superficial...like hair color, eye color, height, maybe temperament, etc. 'Character' speaks to more deep, meaningful traits of a person, or here, a generation of people. Things like what a person values, how they carry themselves in the world, how they treat others, or, here, that they value being loved more than pursuing greatness. Using 'character' also sounds better, in my opinion.

Character is far deeper than a mere trait, or characteristic. So, it depends on what you want to express.

I would write "the tendency".

"Character" is the better word to use here. "Seeking to be loved rather...." is a "characteristic" of people who have such "character". You could use "characteristic", but you would first need to change the sentence like this: "This indicates A CHARACTERISTIC of modern youth, WHICH IS TO seek to be loved rather than to become great. Merely changing "the character" to "a characteristic" would produce a good sentence but that sentence would not specify which characteristic you are talking about. In other words, in that sentence "this" would have no referent in that sentence.