Who are the Siddiqūn, what are their characteristics and virtues, and how does one become one of them?

High aspiration and seeking noble matters are permissible and desirable, and there is no harm for a Muslim to supplicate to Allah to be among the righteous awliya' (saints/friends of Allah) or the sincere Siddiqin (truthful ones).

The awliya' of Allah are the pious among His creation, and wilayah (sainthood/friendship with Allah) varies according to faith and piety. The highest of its ranks, after prophethood, is the station of Siddiqiyah (truthfulness), and these are those who have perfected the stages of faith, righteous deeds, beneficial knowledge, and sincere certainty.

Whoever follows the Siddiqin in their truthfulness, piety, and righteousness, will be counted among them and gathered with them, for Allah Almighty says: (And whoever obeys Allah and the Messenger - those will be with the ones upon whom Allah has bestowed favor of the prophets, the steadfast affirmers of truth [Siddiqin], the martyrs and the righteous. And excellent are those as companions.) An-Nisa'/ 69.

So, whoever desires this high station, must adhere to complete truthfulness in speech and action, and to the fear of Allah (taqwa (definition)) in public and in private.