Let's look at what an autobiography is so that we can decide how to attempt one from a brook's point of view. An autobiography is a self-written story of a person's life. Knowing this, we have to assume that your assignment has to be written from the brook's point of view in the poem "The Brook." Therefore, the brook MUST be the speaker of the poem. Further, an autobiography is usually told in chronological order, so get ready to tell the story from the brook's point of view from its creation to the present day.
The key of the brook's creation is the very first line "I come from haunts of coot and hern." I suppose the brook could consider its creation to be continual if it "comes from" a place where "coot" and "hern" reside. Perhaps it originates from a spring? Feel free to choose a source, just for fun. However, keep in mind that all bodies of water generally go from smaller to larger (and eventually wind up in the ocean). We see this in the following line, that you could use: "I chatter, chatter, as I flow / to join the brimming river." If you would like, you can go to a time of greater earthquakes and dinosaurs, and have the brook speak about its actual creation in that regard.
As most of the poem is about the brook's present day life, that is where your autobiography assignment should focus. You can have the brook talk about the "curves" in its "banks" these days as well as how it feels flowing over "pebbles" and by "men." You can have the brook talk about different plants if flows past such as "willow-weed" and "mallow" and "forget-me-nots." You can have the brook talk about fish and boats and towns. You can have the brook talk about how some places are "shallows" and some are deep. These are all things the brook would impress upon a reader as things that are in the present tense, but will continue.
As a hit to the future, make sure the brook speaks about its everlasting life in the repeated line: "But I go on for ever."