Al Khums is 130 km east of Tripoli along the coastal highway. The drive takes about 90 minutes in light traffic. Shared taxis (servees) and minibuses run the route regularly from Tripoli's main bus and taxi stands and are the standard way to travel for Libyans. The fare is very cheap by any standard.
There is no train service. For visitors coming from Tripoli, the most practical option is to arrange a taxi for the day — the driver can wait at Leptis Magna while you explore, and this also makes it possible to stop at the ruins of Sabratha on the way back. Tripoli's formal taxi services can arrange this.
Note: visitors to Libya require a valid visa, and the security situation has been variable in recent years. Check current conditions with your government's travel advisory before planning a visit.