I hedged on whether to say "music writing" or "writing about music" above. The whole activity is fraught, from the name onward. "Music writing," because of its very laxity, is more encompassing than "writing about music." Much music writing is not necessarily about music, but about the production of or personalities behind the music, or of the cultural or social contexts of the music. Writing about music as such can fall into several traps. Think of an axis with two poles, descriptive and interpretive. Now imagine a perpendicular axis ranging from technical or impressionistic. Technical description or interpretation of music is beyond my capabilities to write or to fully understand. I'm stuck in the camp of pure impressionism--the subjective response to the music, to what it makes me feel or think of. This is the heart of criticism, but can lead to writing that is murky at best and solipsistic at worst.
With popular music, especially in the case of a songwriter like Dylan, there's a further split between talking about music or talking about lyrics. With some technical prowess, one might talk about the relationship of words and music, but in the case of Dylan the words have always been paramount. (The lyrics have been published without music on several occasions.) I number myself among ham-handed lyrical exegete of song lyrics, that often resort to gobbling quotations into the vague engines of interpretive prose. Here's a snippet from Mark Polizzotti''s Highway 61 Revisited stays away from the worst of this, though he does have a penchant for lyrical allusion--more than one ending is called as a "restless farewell." Polizzotti's short book on Highway 61 Revisited is in the attractive 33 1/3 series. Each book in this series pays tribute a classic album in one way or another. A variety of approaches are used, including fictional, descriptive reporting, and subjective impressionism. Polizzotti's book is largely descriptive of immediate context of the music's production and of the music itself (doing a nice job with the latter). As mentioned, he does some lyrical interpretation, mostly in identifying who the various songs may have been addressed to. In doing so, he makes Dylan's songs all seem like diatribes, like complaints against sundry individuals. That's true to an extent, but misses some of the songs' richness.
With popular music, you're probably better of working with context--how the work of the artist positions itself within a subculture, say, or how the culture at large reads it. The organization of Ellen Willis's gives us some guidance here: after an opening section of straightforward aesthetic criticism, there's a section on the context of fandom, one on the sixties, one of feminism, one called "The Navigator" that includes economic and geographic perspectives, and one that engages social class. Not that this was planned; this book collects columns from the 60s and 70s responding to what was actually going on in the world of pop music at the time. Willis, the first pop music critic of The New Yorker, wrote about a now fairly predictable swath of classic rock through early punk--Dylan, Velvets, CCR, Janis Joplin, New York Dolls. She was the first to notice Dylan's manipulation of his own persona--the artist providing context for himself. Between Polizzotti and especially Willis, I've come to understand Dylan much more as a man of his times, responding to his audience's responses to him as much as anything. That's not just self-obsession, since it invokes an underlying struggling for self-determination that was reaching a new high pitch in the 60s.
Good music writing is out there. I always read the current pop critic in The New Yorker, Sasha Frere Jones, and I read Matthew Perpetua's great Fluxblog on a regular basis too. There's still a lot of bad music writing too, but you can find that on your own. Since I'm not liable to be on the subject of writing about Dylan again anytime soon, I thought I might mention a couple of other titles worth looking at. First, there's Greil Marcus's Old Weird America. This, in Marcus's rhapsodic prose, traces the lineage between Harry Smith's Anthology of American Folk Music and The Basement Tapes. Anyone interested in Dylan should also read his memoirs, Chronicles. The first section, on the early sixties, is especially interesting for thinking about his folk period in a beat context.