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What is the typical timeline for going big law after a federal clerkship?

Situation specific - what about a 2 year clerkship for 2018-2020? 2020 seems far away but this upcoming OCI season firms are looking at summer associates that would potentially start in 2020 so I don't know when the right time to apply is.

How do you go about applying? Find a recruiter? Send cover letters to firms blindly? What about if you are clerking in a different state from where you want to end up/took the bar exam?

You generally apply for a post-clerkship job no earlier than about a year before you want to start.
Depending on the market and the court you're clerking on, recruiting season will likely start no earlier that December/January (at least for some of the most competitive firms in the NY market) to start in the subsequent fall, and run through at least May.

As far as how to apply, many recruiters won't work with law clerks, and most law clerks won't work with a recruiter. (Why would a firm want to pay a clerkship bonus and a recruiter fee?) Depending on the location of your clerkship, there may be a well-established recruiting season where firms come and recruit law clerks. If a firm you're interested in isn't coming to recruit, then just send an application to the recruiting contact on the firm's website.

Finally, a caveat: If you were qualified for big law before the clerkship and/or if you had a summer associate position, you'll most likely be fine getting a big law job. But if you weren't and you didn't, it'll be more of an uphill battle, and starting early and networking will both be beneficial.