The sketch of any character's traits in any play is derived from the dialogue. Direct characterization comes from the character's lines, while indirect characterization comes from other characters' lines and reactions. Duke Senior has an interesting role because he serves as the philosophical voice that mediates different perspectives.
His first speech in Act II, scene i, orients the audience to his personality and his philosophy. He starts by praising the pastoral ideal by saying,
Hath not old custom made this life more sweet
Than that of painted pomp?
He then goes on the explain his philosophy that adversity can prove to be a crowning jewel instead of venom:
Sweet are the uses of adversity,
Which, like the toad, ugly and venomous,
Wears yet a precious jewel in his head;
In other speeches, for instance with Orlando and Jacques, he draws out then measures and evaluates the thoughts of others; and as well, he is the distributor of truth and justice.
And all the embossed sores and headed evils,
That thou with licence of free foot hast caught,
Wouldst thou disgorge into the general world.