Was not planning on going, but we are now interested in going to Wimbledon tomorrow (opening day) or Tuesday. Last minute and obviously we will need to queue - do we need to be there by ~5am? I know people camp out...

If we do get there around 5/6am, what time can we expect to actually enter the grounds? Only interested in grounds passes, not court 1 tickets.
Also, is the tube the easiest and best way to get out there from London (Chelsea/Kensington area).

Tube to Southfields then bus or walk to the grounds is the best way. Choose a dry day for ground tickets otherwise you won't have a fun day.

A bit late to answer (as presumably you have already gone or not), but my experience from 2008. I showed up at 7:45AM I still was easily in the grounds. If you want to get into Centre Court, or court 1 or 2, you need to camp, but if you just want to get into the grounds, showing up by 8AM was fine. They don't start letting people in until around 9:30, and the action doesn't start until noon, so you might have some time to kill, but hospitality is all open and so you can get some strawberries and cream and a Pimm's and chill out for a bit. If you do want to get to centre court, there will be points later in the afternoon where they will offer tickets for people already in the grounds as resale.

I wrote about my experience here. Details from Wimbledon itself can be found here.