From Napoli "Piazza Garibaldi" train station:
The trip needs two steps using Circumvesiviana trains from Neaples to Sorrento then using SITA buses from Sorrent to Amalfi. The Circumvesiviana trains, from Piazza Garibaldi station to Sorrento take 50 minutes around with departures every 20 minutes starting from 05.00 AM to 11.00 PM. 

From Capodichino airport of Napoli:
Use taxi or airport bus to Napoli "Piazza Garibaldi" train station then follow the previous instructions.

From Salerno using SITA line buses
Use trains from Roma Termini station to reach Salerno Station then use SITA line buses from Salerno to Amalfi.

Click here to get the buses timetable from Salerno to Amalfi.

Map from Salerno station to SITA buses stop directed to Maiori