It has since become a contract-to-hire with an undefined length of time. [...]
When you are working as a contractor, it is to your advantage to have a well-defined time limit to your contract. Contract renewal is the perfect time to negotiate a rate increase. (If you have agreed to an "undefined" length of time – did your original contract actually lapse and you kept on working? or is it really of no specific length? – then it isn't as clear-cut.) Give your client some advance notice of your intent to negotiate a rate increase upon renewal.
As to the frequency, I would restrict rate increases to at most once per year. It is reasonable for you to expect an increase once per year as that's typically the frequency for budgeting and employee raises. (But always keep in mind: You are not an employee.)
If you ask for increases more frequently than once per year, you will annoy your client. Trust me on this. (If you live in a country with hyperinflation, then you can cautiously disregard this point.)
So, if you are up for renewal every three months (for instance), you won't be asking for an increase at each of those renewal points – only the one that coincides with a year since your previous rate increase.
However, once per year is a guideline for the minimum. I've been in a situations where the client's business was being adversely affected by economic conditions (e.g. 2008-2009), and I postponed rate increases accordingly. Seek to understand the health and profitability of your client's business. For instance, coming out of a recession, did your client just have a good quarter and issue bonuses to employees? That could be a good time to ask for your increase, if you had been delaying.
BTW, I've had one client tell me before that "we don't give increases to contractors." From what I could tell, this client had a revolving-door of contractors who rarely stayed on for more than nine months. At that point I had been there one year and they wanted to renew me for another six months. When I said I'd prefer to conclude the arrangement to find something with a rate closer to my new value, they changed their mind. The only concession the client asked for (and got) was for me to hold my rate constant for one year from the increase – something I was planning on doing anyway!
And yes, it is reasonable for you to expect your rate to be increased. The cost of living is going up all the time, as should be your skills & value. You should aim to increase your rate both by a percentage representing inflation (to maintain your purchasing power), as well as some amount to reflect added skills, experience, responsibilities, or other special added value you're bringing to your client.
Be prepared to reflect on your accomplishments when you talk about a rate increase. Be awesome and consistently impress your client and they shouldn't have a problem agreeing to reasonable annual rate increases that keep your rate moving ahead of inflation.