It's not clear exactly what your first sentence is asking for (there are a variety of surveys and books). For instance, are you only interested in finite Coxeter groups?
The embeddings of the noncrystallographic Coxeter groups $H_3, H_4$ indicated in the follow-up question probably originate in older work of Coxeter and du Val (though I don't recall the exact source). These embeddings have been studied over the years in different styles, but a single comprehensive reference may not exist. One direction is indicated in a series of papers by Matthew Dyer, e.g., Embeddings of root systems. I. Root systems over commutative rings, J. Algebra 321 (2009), no. 11, 3226–3248.
Another direction of research connects the embeddings with mathematical physics, sometimes in rather a computational style, e.g., Mehmet Koca, Ramazan Koç, Muataz Al-Barwani, Noncrystallographic Coxeter group $H_4$ in $E_8$,
J. Phys. A 34 (2001), no. 50, 11201–11213.
ADDED: Older references for the embeddings were given in the notes to Section 2.13 in my 1990 book Reflection Groups and Coxeter Groups, though in the published version I wrote $D_3$ where $D_6$ is intended (this and other corrections have been maintained on my homepage).
NOTE ON HISTORY: Having looked further into the tangled literature (which reinforces my decision not to become a historian of contemporary mathematics), I would replace the "probably" in my second paragraph by "possibly" given the lack of documentation.
(1) The earliest relevant paper seems to be the short note (in an obscure Japanese journal): Jiro Sekiguchi, Tamaki Yano, A note on the Coxeter group of type $H_3$., Sci. Rep. Saitama Univ. Ser. A 9 (1979), no. 2, 33-44. This follows their note on $F_4$ and both are motivated especially by invariant theory aspects. At this remove at least one author recalls they also discussed $H_4$ at that time but didn't write down the details. (2) In the short remark 3.9(b) of his 1983 paper linked above, Lusztig applied some of the previous Hecke algebra and $W$-graph development to the embedding of $H_4$ into $E_8$. His proof comebines somewhat sophisticated previous steps. (3) The most comprehensive treatment of embeddings of non-crystallogaphic finite Coxeter groups is given more expansively in $\S2$ of a Russian paper, with English translation: O.P. Shcherbak, Wavefronts and reflection groups, Russian Math. Surveys 43:3 (1988), 149-194. See Theorem 2 and the diagrams there for $H_2$ (dihedral of order 10), $H_3, H_4$ as well as comments on embedding other non-crystallographic dihedral groups. His method is entirely different from Lusztig's, whose work he was apparently unaware of; he does acknowledge the note by Sekiguchi-Yano. Shcherbak's work is in the mode of the Arnol'd-Brieskorn work on singularities and extends it to non-crystallographic cases.
While I referred to (1) and (3) in my 1990 book, I overlooked Lusztig's remark at the time even though his paper is in the reference list. He and Shcherbak arrived (I think independently) at the same embedding via a "folding" diagram for $H_4$, but their methods are totally different. Scherbak's approach may be more self-contained (assuming one can access the article or translation through a library). For example, he constructs a transparent homomorphism from $H_4$ into $E_8$. Here one sees the Coxeter diagram edge of $H_4$ labelled 5 as arising from a Coxeter element of order 5 in an $A_4$ subgraph of $E_8$. (Some of Scherbak's earlier work on $H_4$ appeared only in a non-translated short note in the same Russian journal in 1984.) It is this work which figures in later developments by B. Dubrovin and others on classification of 2D topological field theories, etc.