There are essentially two ways into Borj Mokhtar: a flight from the north, or a very long drive across the Sahara. There is no third option, and neither is comfortable.
By air. Bordj Badji Mokhtar Airport (IATA: BMW) has occasional Air Algérie and Tassili Airlines flights from Adrar, with onward connections to Algiers. The schedule is sparse — sometimes one or two flights a week — and cancellations are routine. Book early and stay flexible.
By road from the north. From Adrar the route runs south through Reggane and then a long, mostly empty 650 km piste to Borj Mokhtar. The road is tarmac in stretches and graded sand in others; allow two long days in a 4WD, more if you stop to wait out wind. Fuel is available in Reggane and again in Borj Mokhtar but not in between. Carry food, water, a satellite phone, and spares.
By road from the south. Coming up from Gao in Mali, the desert track climbs through Tessalit to the Malian border post and then on to Borj Mokhtar. This crossing has been closed for years at a time, so confirm before you set out.