I've managed to do this at three sites (PKR, Full Tilt, Stars) and ran the
888 $8 sign-up bonus up- I've never actually deposited on any site, but have managed to play tens of thousands of
hands at NL10. In each case my strategy was the same- use freerolls to get to something like $8, then shortstack NL2 full ring, leaving the table when I doubled up, then moving to full stack NL2 at about $40. It was a lot easier when I was doing this (PKR in something like 2007, Full Tilt in about 2010, 888 after Black Friday) as there were more freerolls which didn't require you to have deposited. I'd echo what some players have said above- you need to:
(1) be super-conservative about bankroll management until you get up to about $50, which is an adequate roll to just grind NL2. I can't emphasise enough how dull it is to be kicked exclusively down to mass-field freerolls after playing cash for a bit (I bounced in and out of playing cash a couple of times at FTP before getting comfortable)- you can no longer choose exactly when to play, and you lose perks like FPPs. Avoid this by staying as low as you can. Even now, I run BRM of at least 50 BIs for cash, to minimise my risk of ruin.
(2) choose spots carefully and look for overlays and added money in tournies. For example, my roll was boosted by a $20 score in a $10k freeroll on Stars in mid-2013, followed by a $113 2nd place the next month in a $1.1 tourney with $500 added to the price pool. Those two tournaments, and a few others around the time, were what gave me a comfortable roll on Stars to go with NL5 and some $3 buy-in tournaments.
(3) Promotions can help a lot (NB- they're not as generous as they were in the day). The FTP Academy, which ran from 2009 until Black Friday, paid out a bunch of points which I used early on for buy-ins to occasional MTTs to preserve my bankroll. When there are special offers on sites which don't require you to bust your BRM, make use of them.
(4) Patience is key. If you're a good NL2 player, you might make 6bb/ 100- that's around 12c per hour single tabling (and you won't have the roll to multitable initially). It will be a grind to get your roll up.
(5) Don't multitable anything other than freerolls at the start. If you go on tilt, you're on tilt at all tables, and can burn cash quickly.
(6) Don't play Zoom until your roll is solid. I can't emphasise this enough. Winrates are lower, hands come quicker, tilt is more ruinous, reads on villains are harder to acquire.
One final thing- building a bankroll from nothing remains doable imo, but will be hard- much harder than ten years ago, when freerolls with hefty prize pools were frequent, and reward schemes mainly helped winning players. Remember- most poker players lose money over the medium and long term. If you're merely an average player, it won't happen, as the average player loses something like 10bb/100 at NL2- you will have to work on your game and be very disciplined. Good luck.