Its common for new designers to take on volunteer/unpaid projects for the purpose of building a portfolio, practicing new skills, or developing a client base. When the designer has reached a certain level of proficiency and would like to discuss compensation with the same clients that were receiving free work, what is an appropriate approach?
There is no appropriate approach. Clients that convinced you to work for free aren't the kind of clients that have any interest in paying you. Go find new clients.
DA01 is correct; there's almost no way to transition gracefully and resentment-free from pro bono to paid with the same clients. Possibly if you made a big splash about starting your own company, with a suite of stationery and a website and a literal shingle outside your door, and used that as a way to say that you can no longer afford to do volunteer work (but could offer them a discounted rate as a thank-you for their support), but otherwise, yeah, your free clients have to stay free.
The problem is that your relationship with your pro bono client is "I'm doing this for you out of friendship or the love of the work." Once you introduce money into the equation, then it becomes a business transaction, and some people really object to that. I won't say all; I've had friends who were willing to pay me, but they are exceedingly rare. (and, I might add, really good friends.)
Your best bet is to establish new business-only relationships with new clients and try not to let your free clients overwhelm your time.
You also raise the question of 'when' to transition, which is an important aspect to consider.
Four questions that you may want to think about come to my mind:
1) is your portfolio good enough to attract paid work? Do you have evidence of that?
2) does the 'free' work you provide add increasing value to your portfolio?
3) does the 'free' work help you improve your skills?
4) are you able to continue working for free and support yourself from other sources?
If you feel your portfolio is established enough to attract paid customers, then it may be the right time to aim for a transition now. If the free work does however continue to add substantial value to your portfolio then it might be worth holding out for a bit longer. If it doesn't then it may at worst be a waste of your time. Of course you may also be able to improve your skills, which may be a reasons for continuing to offer free work. And finally you may need to consider the practical and financial implications of working for free. It's best to seek paid work opportunities before you end up being desperate for money. It's therefore useful to think ahead, which seems to be what you are doing. Good luck!