Saladin wasn't successful in the Third Crusade. He lost the 3 main battles in the Third Crusade (Acre, Arsuf, Jaffa), and the Crusaders would probably have captured Jerusalem if they had continued and attacked the city. Jerusalem remained in Saladin's hands because the Crusaders were split into two factions: one wanting to capture Jerusalem, and the other (Richard I's faction) wanting to consolidate the territory they had taken along the coast. The coastal territories were important for trade, and essential for maintaining the Crusader states; Jerusalem would have been difficult, if not impossible, to hold, and attempting to hold it might have led to the destruction of a large part of the Crusader army and left the Crusader states too weak to hold the coast. Thus the conservative, and possibly correct, decision to turn back from Jerusalem.
OTOH, Saladin was very successful leading up to the Third Crusade, and it was this success that led to the Third Crusade, following his victory at Hattin and his capture of Jerusalem. Saladin's success was due to (a) luck, (b) political skill, and (c) military competence.
(a) Luck
Saladin's successes depended on his control of Egypt. Egypt gave him a strong economic base (from both trade and Egypt's resources), and allowed him to build and finance the army that took Syria.
Saladin's control of Egypt was helped by a series of lucky event. His uncle Shirkuh was sent by Nur ad-Din Zengi, the ruler of Syria, to intervene in a civil war in Egypt to support Shawar, who had defected from being a Fatimid governor of Egypt to becoming a Syrian governor of Egypt. Following Shawar's re-instatement, a 3-way war broke out between Shawar, Shirkuh, and the Crusader kingdom of Jerusalem, which resulted in Shirkuh controlling Egypt, the Crusaders expelled from Egypt, and Shawar's execution. Two months later, Shirkuh died (of illness). Saladin became the new vizier, subject to Nur ad-Din.
(b) Politics
Saladin was able to retain control of Egypt through political and administrative skill. Saladin was a popular ruler in Egypt due to his support of Sunnism (the majority of Egyptians were Sunni, and had been under the rule of the Shiite Fatimids), his policies of religious tolerance (which won support from Christians and Jews), and tax reform. Saladin was able to overthrow the Fatimid Dynasty (he had been ruling as vizier nominally on behalf of the last Fatimid Caliph, but in practice on behalf of Nur ad-Din) by waiting for the Caliph to die (more luck, assuming that the Caliph died of natural causes - Saladin was rumoured to have had a hand in it).
Saladin had a longer-term problem: Nur ad-Din. Nur ad-Din found the tribute payments from Egypt unsatisfactory, and was probably planning to replace Saladin. Saladin postponed the coming confrontation by evasiveness and delaying, but once Nur ad-Din finished his other, more urgent wars, Saladin and Nur ad-Din gathered their armies for the struggle. At which point Nur ad-Din died (of illness - more luck for Saladin). While Nur ad-Din had been a strong and effective leader, there was no strong and effective successor - his 11 year old son became the new leader in name, and civil war broke out in Syria between the various claimants (including Saladin) for the regency and guardianship of Nur ad-Din's son. Saladin's strong economic base, together with military competence, allowed him to win that civil war, and become the ruler of Syria as well as Egypt.
The control of both Egypt and Syria left Saladin with a strong base for fighting the Crusader states (which led to the conquest of Jerusalem) and attempting to control Mesopotamia (which he did not achieve). War is an expensive activity, and Saladin's successes depended very much on his economic resources.
(c) Military competence
There was a lot of fighting in the events outlined above. Saladin was successful. In some battles, mere avoidance of incompetence should have been sufficient, since his army was strong. However, he was successful even in closer battles. One example is the Battle of the Horns of Hama, 1175, where he defeated a large Zengid army and took control of Syria. Saladin deployed in a strong defensive position, and waited for the Zengids to attack. They attacked, and Saladin was hard-pressed, but the arrival of Saladin's Egyptian reinforcements turned the battle decisively in his favour.
For an excellent biography of Saladin, I recommend Anne-Marie Edde, Saladin, Harvard University Press, 2014.