The horoscope of Lord Rama is presented in the article on Rama and Krishna. According to Ramayana, the date of birth of Rama is the Shukla Paksha Navami of Chaitra Masa. Unfortunately, there is no mention of any year or a reference to a date in present time through which an unambiguous claim can be made. Even more unfortunately, unambiguous in Kaliyuga means a reference to dates within the western calendar.
What is mentioned in the epic is the planetary combination and location of planets in various signs and houses at the moment the Lord was born. I came across a website The Historical Rama, which attempts to solve this puzzle of finding out a date where this planetary combination was possible. It uses a planetarium software (but does not mention which specific one) and arrives at 12:30 PM on 10th of January, 5114 BCE as the date and time of birth. Precession of Equinoxes takes care of moving the Ram Navami, which usually falls in April in present times, to January, in about 7000 years in the past.
It is a good attempt and I hope it helps to put an end to the discussion raging in our country on Rama and His existence. I was hoping it would clarify the position of Mercury in His chart – where I maintain that it is in eleventh house and some of the readers are of the opinion that there is a Budhaditya Yoga in the chart.
It would have been wonderful if it had some way of putting an end to the controversy of the place of birth as well.
Postscript as on May 4th 2012: The site of Historical Rama is not active anymore. The domain expired some time back and the owners did not bother to renew the domain. It is a nothing but a travesty… However, thanks to Wayback machine, it is possible to look at some expired pages. I have replaced the link with the Wayback link. The dates of various events is mentioned in this page – http://web.archive.org/web/20100609171913/http://historicalrama.org/dateevents.html. There is another site http://historicalrama.com/, but I am not sure if this is a 100% copy.
There is another site http://bharathgyan.com/ that has published a book on this topic – worth visiting. It maintains some articles on the site, but not as extensive as the main site that does not exist anymore.
Postscript on September 19th 2013: This is a good article on Speaking Tree about LORD RAMA MYTH OR HISTORICAL SUPER HERO. Please go through it. Very interesting and very informative.