The word has rather a nice sound today: anyone who is doing something "doggedly" is surely to be praised for not giving up. But this sense has been around only since the mid-18th century. The original use, from the 14th century, expressed the fiercer canine qualities. Shakespeare uses the word just three times. In King John, war is described as dogged (IV.iii.149), and the sense is plainly "cruel, ferocious". Earlier in the play, Hubert reassures young Arthur, "I'll fill these dogged spies with false reports" (IV.i.128). He is not praising the spies for their tenacity; they are Arthur's enemies. And similarly, when Gloucester describes the Duke of York to the King as "dogged York, that reaches at the moon" (Henry VI, Part 2, III.i.158), he is thinking of him as an ambitious malevolent schemer. The relevant meaning is "spiteful, malicious".
David Crystal is author, with Ben Crystal, of Shakespeare's Words, published by Penguin