There is a direct connection from Athens by airplane (45 minutes) and by boat from Piraeus. You can also use take an airplane to neighboring islands, and then reach Leros by catamaran, boat or other airplane.
Airlines: Olympic Air www.olympicair.com
Ships: www.ferries.gr, www.gtp.gr, www.openseas.gr, Blue Star Ferries (www.bluestarferries.com)
Information about Catamaran http://www.12ne.gr/en/
Information about Patmos Star http://www.greeklodgings.gr/eng/patmos/ferry.html
Information about Nisos Kalimnos Ferry Boat http://www.anekalymnou.gr