Hi there Chris. I'll answer your question real quick, and then give you some suggestions on what you could do based on my own experience. Unless you have 3-4 years of experience in both Latin and Greek with a high GPA, you will have an extremely difficult time getting into a Ph.D program in classics (and by extremely difficult, I mean the chances are close to nil...if you have 2 years of both, you may have a slight chance).
I too graduated with a BA in history from a school that had no formal classics department. I had 3 semesters of Latin and 2 semesters of Greek...which was just barely enough to get me into UPenn's post-bac. The post-bac has done a wonderful job catching me up with my languages and teaching me the finer points of Classics. Since I was a general history major...I was still surprisingly ignorant of the Classics despite my own reading on the subject.
It sounds to me that you need to look into post-bac programs and terminal MA programs. It will be the most efficient way to get caught up on those languages (unfortunately reading Latin on your own doesn't count for much to graduate committees...they want to see a GPA and specific coursework). I can vouch for Penn's post-bac. It is very difficult, but the professors whip you into shape, and if you perform well, they will write you a good recommendation letter. Other post-bacs you can consider include: UCLA, Georgetown, Columbia, and UNC. I believe any one of these would set you on the right track (unfortunately they are all unfunded...Penn's costs about 10,000 for tuition).
You should also look into terminal MA programs. Many of them are guaranteed funding:
WashU in St. Louis
Notre Dame
Vanderbilt
These are all well funded with a stipend...which means they are almost as competitive as the Ph.D programs! I know Vanderbilt only accepts 2 or 3. Some are not guaranteed funding, but are worth applying to, ie. University of Georgia. If I were you, I would apply to 5 or 6 MA programs and 3 post-bac programs. If you apply to Ph.D programs, you will be competing against applicants with an MA and 4-5 years of experience in both languages.
Best wishes!
Andrew