There are only two significant Greek Byzantine sources for the events of the First Crusade:  Anna Comnena (Komnena) and Zonaras.  Anna, daughter of the emperor Alexios is absolutely fundamental.  Zonaras, provides only a brief epitome, with little new information.  

Anna Komnena

{1081-1118}

[CE 1:63] (wiki)  

Greek: 

  1. (1) Leib, B. (ed. and French trans) Alexiade: Regne de I'empereur Alexis I Comnene (1081-1118), 3 vols. (Paris: Societe d'edition les belles lettres, 1937-1976) (critical edition with translation and commentary in French)
  2. (2) Greek text online

English:  

  1.  Elizabeth Dawes (trans.), The Alexiad,  [available online and kindle]

           Online version of Dawes

     2. E. R. A. Sewter (trans.), The Alexiad of Anna Comnena, 2nd ed. (Harmondsworth: Penguin, 2009) [available in •paperback and •kindle]

Zonaras

{1081-1118}

[ODB 3:2229] (wiki)

Greek: 

  1. Zonaras, John, Annales, ed. Moritz Pinder (CSHB, 2 vols.), and vol. 3, ed. Theodor Biittner-Wobst. (Critical edition)

     Online version of volume 3.  

  • Only volume 3 contains discussion of the era of the Crusades
    • Romanus IV Diogenes {1068-1071} = Zonaras 18.10-15 (3:682-707)
    • Michael VII Doukas {1067-1078} = Zonaras 18.16-17 (3:707-717)
    • Nikeporos III Botaneiates {1078-1081} = Zonaras 18.18-19 (3:717-726)
    • Alexios I Komnenos {1081-1118} = Zonaras 18.20-30 (3:726-768)

English: –NA for the period of the crusades

      parallel Latin translation is found in (1) above.