I have played this scenario fairly extensively over the past few months, from all three sides. These are my observations:
Two basic points to note: First, the scenario departs from historical events in that Didius Julianus, in all my playthroughs, is not murdered, with Italy subsequently falling to Septimius Severus fairly quickly. Rather, Didius must be labouriously defeated. Second, Pescennius Niger has by far the strongest land army. However, in all my playthroughs as Septimius Severus or Clodius Albinus, Niger remains passive almost until the end, not doing anything beyond engagement in minor skirmishes. His main army does not move west until it is too late. This roughly corresponds to historical events; Niger acted too late against Septimius Severus and then lost key battles.
As to the three campaigns:
1. Septimius Severus: What has worked best for me is to split my army into three - one main force to head west and take rome before Clodius Albinus can, one smaller force to stay east and engage the westernmost elements of Niger's forces before they can cause trouble in Thrace, Macedonia and Greece, and a small force to defend the Danube and Dacia. As to the latter, watch for a message telling you how large this force must be to avoid a resurgence of the Marcomannian wars. As soon as you have taken Rome, or even earlier, take on Clodius Albinus. The later you declare war against him, the stronger he will become militarily, leading to a drawn-out war that may cost you dearly when Niger does decide to go west. You will not be able to hold of Niger while at the same time fighting Albinus in the west. Even if Niger remains in Syria, an early war against Albinus seems preferrable.
2. Pescennius Niger: The most straightforward campaign. What has worked best for me is to gather my armies and move west promptly. This helps, first, to bind Severus's forces in Thrace and along the Danube, making it more difficult for him to succeed against Didius Julianus in Italy and, eventually, once their alliance breaks down, against Clodius Albinus. Second, it is feasible to move into Greece and launch a naval assault against Rome straight from there, defeating Didius Julianus and making Italy and North Africa flip to Pescennius side. From there, Pescennius's superior military might makes successful wars against Septimius Severus and Clodius Albinus winnable.
3. Clodius Albinus: The most difficult campaign. I have won this only once. In that playthrough, Septimius struggled against Didius Julianus, never making inroads far into Italy and camaigning rather unsuccessfully against Pescennius Niger further east. This allowed me to defeat Didius Julianus and make Italy join me. I then managed to wear Septimius Severus's forces down, and Pescennius Niger simply never engaged me in a major war at all, allowing me to occupy and hold all my objective cities. Playing as Clodius Albinus, the overall size of the military force at your disposal, either from the start of the scenario or by recruitment later on, is very limited, and it is quite key to avoid major losses through defeats or attrition.
Hope this helps. Would be interested to hear how others have found this scenario.