Welcome!
I started with a £20 deposit about two and a half years ago.
Like you I've a family and have pretty limited time to play, usually an hour or so when the kids (and sometimes the wife) have gone to bed.
10 hours play would be a good week for me.
I am $14,000 up at the moment and in ring games my hourly win rate is around $30-40/hr.
You can definitely have an enjoyable and rewarding hobby doing this, but the critical thing is to do it right. That means making sure you are properly bankrolled for the limits you play (so you have no realistic risk of ruin), no cashing out (except for emergencies) until you achieve a big enough roll to play the games you want (in m,y case that was $8000 for the $200 NL tables), always striving to improve and lots and lots of discipline...
Post any hands that give you trouble in the appropriate section of this forum so people can help you, and get Poker Tracker as soon as you can afford to (I'd suggest as soon as your roll reaches $200 or so)
Good Luck...