You can climb it in 4 pitches but most seem to break it into 6-7 shorter ones given the meandering nature of the ridge. I've seen parties try to climb it using the full length of the rope and they get hung up on various features. Also, I think the grade varies widely depending on which way you decide to pass the slower parties.
There are two basic ways to approach the Chamonix Valley.
From the West (and in winter): Fly or drive into Geneva and then take a train or bus into Chamonix which is one hour away. Once in Chamonix, the Chamonix Bus runs all day and until 9PM stopping at the various points downtown and the lifts leading to various climbing objectives. To get to the Index, take the bus or drive (direction Argentiere) the 2 KM east to La Praz and park at the Flegere lift. Take the lift up to the top station (Index) and walk 10 minutes to the large index finger shaped formation right next to the lift.
From the east (warm months only): Drive from Bern through Montreux along Lake Geneva until you get to Martigny, Switzerland. From there, follow the well signed road 50 KM to Chamonix. This is the more scenic road to take if you have the time. The Flegere lift is on the right in La Praz 2 KM before you enter Chamonix proper.
The SE ridge is actually 4 pitches of III with a couple of well protected moves of IV on the first pitch.
The index is a very nice mixed climb in winter. You have to take care on the traverse of the lower snow field. Bring Ice Axe and crampons.
Also remember that much of the fixed gear is hidden under the snow.