Most accounts of Spartacus indicate he was at some point a Roman soldier, which helps explains his tactical abilities.

A lot of the revolt's success was based on the ability to attract new recruits by demonstrating their ability to beat Romans in the field. One of the most notable victories was against Gaius Claudius Glaber at Mount Vesuvius. Spartacus's men were besieged on Mount Vesuvius, but were able to defeat the Roman militia by climbing down the mountain with vines and attacking the army from the rear. This and other early victories gave Spartacus more weapons and equipment and also inspired more slaves to join his forces.

It's important to understand that the initial Roman response to Spartacus was fairly weak. Rome's armies were occupied in Spain and Anatolia, and, as Appian writes, Spartacus was first countered "not with regular armies, but with forces picked up in haste and at random, for the Romans did not consider this a war as yet, but a raid, something like an outbreak of robbery."

In terms of how he made his fighting force so effective, there are many descriptions of Spartacus emphasizing taking/making weapons and equipment over acquiring gold and silver in order to keep his men well-equipped. He also encouraged his men to fight harder through actions like crucifying a Roman prisoner to let his soldiers know what would happen if they were defeated.

In terms of actual tactics and strategies, it's difficult to find many specific details. Climbing down a mountain and attacking from a rear is the most popular example, but it most writers allude to unorthodox tactics and strategies without actually describing them. All of my research has concluded that the burning logs in Kubrick's Spartacus, for example, weren't actually used. I'd be very interested in hearing specific examples.

In short, the victories in the Third Servile War were due to a charismatic and talented commander who was able to inspire and equip troops while drawing from an enormous pool of potential soldier fighting against an unprepared military that underestimated Spartacus's capabilities. After a few losses the experienced Crassus and his men had a number of victories against Spartacus, and as Pompey returned from Spain and closed in on Spartacus, the war ended pretty quickly. It's not my place to speculate on how well Spartacus would have fared if so much of the Roman military hadn't been so far away from Italy, but I feel it's right to characterize Spartacus's successes as being the right man at the right time.

I'll start by qualifying your original question a bit. Spartacus wasn't an existential threat to the Roman Empire in as much as he showed any sign of wishing to conquer it; I would argue that such threat as he posed was more in terms of undermining the slavery-based economic system. His successes threatened (and, if he had not been defeated, would have perhaps demonstrated to catastrophic ends) to Rome and its slaves that it was possible for slaves to rise against their masters. Appian and Florus mention that Spartacus intended to march on Rome, but he did not do so, and it certainly smacks of romanticism; there is very little other discussion from contemporary sources of any motive apart from an act of rebellion by desperate men, and attempts to simply move beyond the grasp of the Romans, whether by crossing the Alps or by hiring ships to cross to Sicily, perhaps to continue the fighting/banditry/rebellion there among a large slave population. In any case, it's difficult to determine precisely what Spartacus ever intended, and thus to what degree he posed a threat.

In terms of how he achieved the successes that he did, it's important to understand that Rome generally did not keep large forces in Italy proper during the late Republic. At the time of Spartacus's revolt, the bulk of Rome's soldiers were already engaged in fighting in Spain against Quintus Sertorius (the last vestiges of the Marian-Sullan civil war) and in the east against Mithridates of Pontus. The first fights that Spartacus had with Romans were with militia and under-prepared units who certainly made the mistake of underestimating Spartacus. I'm afraid I am not the best candidate for answering your question about the specific tactics he used except to point out that he is supposed to have used ropes to sneak men down Vesuvius when they were blockaded there, attacking Glaber's unfortified camp from behind and taking him by surprise. By the time that Spartacus's men were fighting consular legions, their earlier success had inspired many to join his army, and a pair of legions were fighting tens of thousands of slaves -- note that the legions had major success wiping out a group reported at 30,000 under Crixus before being defeated by Spartacus.