To my knowledge, none of the letters supposedly exchanged between Saladin and Richard I survive to this day. However, one of our best sources for the Third Crusade is chronicle evidence, supplied from both the Latin and Arabic traditions. Some of these have been (mercifully) translated into English through the Crusade Texts in Translation series, published by Ashgate. Of particular interest for the Third Crusade is Ibn al-Athir's chronicle الكامل في التاريخ (The Complete History) and the Itinerarium Peregrinorum et Gesta Regis Ricardi. A couple other authors have also compiled other sources related to this crusade.

D.S. Richards, The Chronicle of Ibn al-Athir for the Crusading Period from al-Kamil fi'l-ta'rikh. Vol. 2. Aldershot: Ashgate, 2007.

Helen J. Nicholson, The Chronicle of the Third Crusade: The Itinerarium Peregrinorum et Gesta Regis Ricardi. Aldershot: Ashgate, 1997.

Peter W. Edbury, The Conquest of Jerusalem and the Third Crusade: Sources in Translation. Aldershot: Ashgate, 1996.

Gabrieli's Arab Historians of the Crusades probably has some useful sources as well, although I cannot remember which ones off the top of my head.

Finally, I would caution readers to be wary of popular histories with regard to the Third Crusade. The relationship between Richard I and Saladin has been sensationalized in the past as a sort of clash of civilizations, which doesn't necessarily hold up under closer scrutiny. James Reston Jr.'s Warriors of God: Richard the Lionheart and Saladin in the Third Crusade is particularly guilty of this, in my opinion. For narrative overviews of the Crusade, I would recommend sticking to Thomas Madden, Jonathan Riley-Smith, Steven Runciman (a bit dated, but still very useful), and Christopher Tyerman. For narratives focusing on the Islamic side of things, try Carole Hillenbrand and Paul Cobb. Hope that is useful!