Fascinating question, Dave, and a challenging one. My Mother's husband is a Levite, a member of the "Priestly" tribe or caste in Hebrew society, so I've always been kinda exposed to Yom Kippur through his side of the family, but I never really looked into it too much until I saw this question.

In answer to your question of how the holiday was celebrated in ancient times, there appears to be surprisingly little floating around the internet to provide such an insight. However, I was able to find a few tidbits of information on Yom Kippur and its observance in ancient times that might qualify.

According to an article posted at History.com:

"Yom Kippur (Heb literally means Day of Atonement) is celebrated on the 10th day of Tishrei, 10 days after Rosh Hashanah. Yom Kippur is first mentioned in Leviticus 23:27 where it is described as a day of atonement when no work should be done. The text goes on to describe the special rites that the priests would do to expiate the sins of the people. One of these rites takes the people's sins and symbolically places them onto an animal which is then driven out into the wilderness. This rite is the origin of the term "scapegoat"."

The above-mentioned act of symbolic purification gives an indication of the original purpose of the holiday, a purpose which still lay at the heart of the holiday even into modern times, though perhaps the methods of obtaining such purification might have changed a bit over the years.

The origins of Yom Kippur, according to Rabbinic Tradition, lay in the time of the Prophet Moses. In an article titled "The Bible Story of Moses, the Hebrew Law Giver: The Childhood and Secular life of Moses" by N.S. Gill, posted at http://ancienthistory.about.com, the author offers the following statement on the origins of Yom Kippur following the destruction of the Golden Calf idol fashioned by the Israelites:

"Yom Kippur

Moses then returned to the mountain where he persuaded an angry God not to destroy, but to forgive his people. When God did so, it became the first Day of Atonement (Yom Kippur). Then, with a second set of divine commandments, Moses again returned to his people from the Mountain-top. "

Thus we see that Yom Kippur, developed during the period of post-Exodus wandering in the wilderness, had its origins in a time when the people celebrating it had no land of their own, and were still very much a tribal society, although led by ecclesiastical chieftains rather than secular ones. One might very well conclude, then, that the methods by which the holiday was celebrated at such a time were very suited to a tribal, nomadic society, such as the ritual condemnation of a goat which is mentioned in the first example listed above.

However, as the Hebrew tribes settled into the region of the Levant, and their means of support altered to a more agricultural footing (land-cultivation and animal husbandry), they built cities, and eventually built a Temple in Jerusalem, their customs relating to the observance of Yom Kippur also appear to have expanded a bit. Here's an illustration of Yom Kippur in the time of the Great Temple of Solomon, courtesy of http://religionfacts.com:

"In the days of the Temple in Jerusalem, the high priest conducted an elaborate sacrificial ceremony on Yom Kippur. Clothed in white linen, he successively confessed his own sins, the sins of priest, and the sins of the people, then entered the Holy of Holies in the Temple - the only day this was allowed - to sprinkle the blood of the sacrifice and offer incense. The priest then sent a goat (the "scapegoat") into the wilderness, where it was driven to its death, to symbolically carry away the sins of Israel."

Again, the goat symbolically condemned with the sins of humanity is present in the ritual, but also there are elements not present previously such as the white linen garments, and the sprinkling of sacrificial blood in the inner-sanctum.

Of course, the customs and rituals described above tend to deal with the role of the Priesthood in observance of Yom Kippur. All my research thus far has led me to conclude that for the vast majority of the Judaic population, the observance of the holiday appears to have changed very little over the centuries. Certain crucial elements are always present, and must be, as they are pre-requisite for the purification which is undergone on the Day of Atonement. According to the Israeli Ministry of Foreign Affairs (http://www.mfa.gov) :

"Yom Kippur, eight days after Rosh Hashanah, is the day of atonement, of Divine judgment, and of “affliction of souls” (Lev. 23:26-32) so that the individual may be cleansed of sins. The only fast day decreed in the Bible, it is a time to enumerate one’s misdeeds and contemplate one’s faults. The Jew is expected, on this day, to pray for forgiveness for sins between man and God and correct his wrongful actions against his fellow man. The major precepts of Yom Kippur - lengthy devotional services and a 25-hour fast - are observed even by much of the otherwise secular population. The level of public solemnity on Yom Kippur surpasses that of any other festival, including Rosh Hashanah. The country comes to a complete halt for 25 hours on this day; places of entertainment are closed, there are no television and radio broadcasts (not even the news), public transport is suspended, and even the roads are completely closed. Yom Kippur in Israel has special meaning due to memories of the 1973 war, a surprise attack launched by Egypt and Syria against Israel on that very day. "

As indicated, the factors of prayer (as in devotional services) and fasting for 25 hours seem to be ever-present, as does the observance of Sabbath by not working on that day. The prayers, it should be mentioned, are very specific ones created especially for Yom Kippur. Religionfacts.com describes the specific Yom Kippur prayers:

"Observances of Yom Kippur

On the eve of Yom Kippur, the Kol Nidre is recited. This prayer is well known for its beautiful melody, but its meaning has sometimes been misunderstood. The Kol Nidre ("all vows") annuls all vows made throughout the year, and Anti-Semites have used the prayer as evidence that Jews are untrustworthy. But the Kol Nidre actually refers only to vows made between oneself and God, and especially frivolous vows made to God when pleading for help or religious vows made under duress (such as professed conversion to Christianity during the Inquisition). {4}

The recitation of the Kol Nidre does not change the fact that obligations towards other people must be upheld. In fact, the eve of Yom Kippur is considered one of the best times to seek and grant forgiveness. God will forgive sins committed against himself, but if one has wronged another person, he must seek forgiveness from that person and try to make it right. The Mishna teaches, "Yom Kippur does not atone until one appeases his neighbors." {5}

On Yom Kippur, Jews must abstain from all work, food, drink (including water) and sex. Orthodox Jews also follow the Talmudic regulations of not wearing leather shoes, not washing, and not "anointing oneself" (i.e., wearing deodorant, lotions, perfumes, etc.).

The majority of Yom Kippur is spent in the synagogue, where special services are conducted from morning to evening. Especially in Orthodox synagogues, it is customary to wear white on Yom Kippur to symbolize purity before God and the forgiveness of sins. Some wear the kittel, the white robe in which the dead are buried.

One special feature of the Yom Kippur synagogue service is the insertion of the confession of sins of the community into the regular Amida blessing. All are recited in the first person plural (e.g., "we have been aggressive, we have been slanderous") to emphasize communal responsibility for sins. {6}

The concluding service is unique to Yom Kippur. Called Ne'ilah, it usually lasts an hour and the ark (where Torah scrolls are kept) remains open throughout the service. Everyone therefore must stand the entire time. The service is the last chance to get in a "good word" before God's judgments are sealed. At nightfall, the Yom Kippur service concludes with one last long blast on the shofar."

It seems quite likely that these observances have been an integral part of Yom Kippur since its inception thousands of years ago, and thus I must conclude, until I'm able to find evidence to the contrary, that, excepting those of the Priesthood whose observances might have expanded throughout the centuries, the vast majority of people who have celebrated Yom Kippur over the years have celebrated it much the same way it was celebrated in ancient times.

Hope that helped a little bit.