By bus, there are daily departures from Buenos Aires to Río Gallegos, from where you can make a connection by bus to El Calafate, and then proceed to El Chaltén. By car, from Buenos Aires via NR 3 to Río Gallegos, then take NR 5 (that connects to NR 40) to El Calafate, and then proceed to El Chaltén along NR 40, taking a detour to the west on Route 23 to finally arrive in El Chaltén. A shorter alternative is to take a gravel road, Route 288, at Piedrabuena towards Tres Lagos, and then take Route 23 to the west to arrive in El Chaltén (verify previously the road condition).
Aerolíneas Argentinas and LAN have daily flights from Buenos Aires, Ushuaia and Bariloche to El Calafate. From there you follow by bus or car to El Chaltén, some 220 km (there are several buses and also car rental options in El Calafate).