Relatively concise English expositions of the proofs of the various Weil conjectures

See Kleiman's essay "Algebraic Cycles and the Weil Conjectures", in the volume "Dix exposes sur la cohomologie des schemas". (Despite the French volume title, the article is in English.) This article focuses on the relation between the Weil conjectures and Grothendieck's standard conjectures, but contains a complete proof of all four Weil conjectures modulo the existence of a well-behaved cohomology theory.

(In fact, the proof, which occupies the last section of the paper, is only about three pages long and self-contained modulo some formal properties of that good cohomology theory, some of which are established earlier in the paper and some of which are conjectural, but that you might be willing to take for granted.)

What about Nick Katz' expose:

Nicholas M. Katz, MR 1831948 $L$-functions and monodromy: four lectures on Weil II, Adv. Math. 160 (2001), no. 1, 81--132.

As well as Kowalski's notes.

The second part of J.S. Milne's Lectures on Étale Cohomology is devoted to the proofs of the Weil conjectures. The theory of étale cohomology is developed in the first part, but if you're comfortable using that as a black box, you can skip straight to the second part (pages 151–200 in the current version).