Night 0 - sleep in Puerto Natales. Disembark at the Park gates. Do not take the shuttle (US$6) to Las Torres, instead hike for free directly from the gates on the trail further from the mountains.
Night 1 - tent at Paine Lake (Campamento Lago Paine or Campamento Coiron on map)
... not on the official hiking map given to you at the Park gates. It is an old campsite, now retired. You can still camp here free but it is not encouraged. To get here means a really LONG hiking day with full pack.
Night 2 - tent at Los Perros (the Dogs)
Night 3 - tent at Paso (the Pass) or Campamento Guardas (both are good)
Night 4 - tent at Campamento Italiano: sidetrip to Frances Valley. Another very long hiking day.
Night 5 - tent at Campamento Torres: sidetrip to Towers Lookout in the afternoon, and/or following morning. Additional sidetrip to Valle del Silencio (Valley of Silence) if time allows.
Night 6 - Hike out to the Park gates, or pay for the US$6 van shuttle, or borrow a voucher from another hiker to save the walk between Las Torres & the park gates. From there, catch your afternoon bus to town. Sleep in Puerto Natales