What does "He was a gentleman on whom I built / An absolute trust. / O worthiest cousin!" mean?

In Act 1, Scene 4, Duncan is discussing the former Thane of Cawdor, a noble who had committed treason against the Scottish throne, with his older son, Malcolm.  For this reason, Duncan ordered the Thane's immediate execution, and he swiftly awarded the traitor's title (and lands) to Macbeth as a reward for his loyal and courageous service to the crown in the battles against the rebel Macdonwald and the Norwegian king's army.  

In the lines just prior to these, Duncan laments the impossibility of knowing what a person is thinking just by looking at his face.  In other words, it is too easy to be deceived by another.  In the lines you cite, he says that the old Thane of Cawdor was a man whom he completely trusted, and so it is that much more upsetting to the king that he was deceived by him.  He then turns to Macbeth, his kinsman, and calls him "worthiest cousin."  This is somewhat ironic because, when Macbeth was told by the Weird Sisters that he would be king, he immediately began to consider what he might have to do (i.e. get rid of the current king) to make this happen.  In other words, Duncan was not only deceived in the old Thane of Cawdor, but he is likewise deceived regarding the trustworthiness of the new.