Is it permissible for a man to approach a girl with the intention of getting to know her and winning her affection, then proposing to her, or is it better for him to focus on building himself up first and then propose?

Not every admiration for a person reflects their true self. For a successful marriage, one must choose a religious wife, consulting others, and then performing Istikhara (prayer for guidance). If the girl is religious and of good character, and you are able to marry and hope for an answer (to your prayer), then propose to her guardian, while observing the Sharia guidelines when speaking with her. Otherwise, be patient until your affairs become easy. Entrust your matter to Allah, supplicating Him to grant you a righteous wife, for what you missed might be better for you, in accordance with His Almighty saying: "And perhaps you dislike a thing and it is good for you; and perhaps you love a thing and it is bad for you. And Allah knows, while you know not."