Not an easy question to answer. Based on the research below, it seems that chairs were reserved for those of high status. That would imply that others sat on the ground/on mats/ or otherwise informally.
Consulting condordance of every occurrence of chair in the Bible. All of these examples imply to me that chairs were reserved for those of status/privilege. (I'm open to counter-arguments; I do not present myself as a Biblical scholar)
aside: @ubiqutous_student points out that the word "table" can also refer to a form of seating. Quick reference to a concordance suggests that the word table is frequent, but refers to the modern table; I suspect that the term to which @ubiquitousStudent refers has been translated as chair. Resolving this would require linguistic skills I lack.
The answer seems to be influenced by technology, economics and culture. In the absence of a conclusive answer, I'd look at the cultures which influenced this era in space & time - Africa, Greece, Rome, etc.
Working from the European culture, chairs were uncommon, which would lead one to assume that they were uncommon in Biblical times.
"In Europe, it was owing in great measure to the Renaissance that the chair ceased to be a privilege of state and became a standard item of furniture for anyone who could afford to buy it. Once the idea of privilege faded the chair speedily came into general use. Almost at once the chair began to change every few years to reflect the fashions of the day." Wikipedia:Chair
On the other hand, in Greek culture, chairs were more common.
"The striking thing about the Greeks," Rybczynski explains, "is that the chairs become very democratic very quickly."
Just take a look at the pictures — "there are women in chairs, gods in chairs, musicians — so it clearly was a tool used by many people," he says. NPR
The most common form of Greek seat was the backless stool, which must have been found in every Greek home Wikipedia:AncientFurniture
Early stools for ceremonial purposes were merely squared blocks of stone. When made of wood, the stool had a flint seat (later shaped concavely) covered with a soft cushion. In time the stool developed into the chair by the addition of a back and arms. Such throne chairs were reserved for use by personages of great importance. Britannica
? What was the role/status of chairs in Rome of the period? My memories are from a later Roman period where chairs were significant indicators of status, but that assumption needs to be tested both for time, and for context.
Rectangular footstools, sometimes with claw feet, were used with the high chairs and couches Britanica
I don't have a source a the moment (nor the time), but in many African cultures, chairs were a significant status indicator.
Which model was expressed in "Biblical times"?
In Biblical times in ancient Israel chairs were not used except in palaces. Joseph's brothers sat on seats at a banquet in Egypt (Genesis 43:33); and David had a seat at the table of King Saul (I Samuel 20:5, 18). BibleHistory
Nothing is known of the form of the chair ("kisse"). It may be assumed that, like the bed, it was similar to the Egyptian, although it may have resembled the small, low stools on which modern Orientals squat in the cafés. In any case chairs were necessary pieces of furniture among the ancient Hebrews, who sat during meals, and did not recline like the Greeks and Romans. JewishEncyclopedia
Note: This is a work in progress; I don't have time for a well researched answer, so I'm copying notes in here.