First of all, Caleb is an orphaned young man who has been taken into and raised under the sponsorship of Mr. Falkland. Falkland moved into an area of England that was under the influence and sway of Mr. Tyrrel. Mr Tyrrel is a tyrannical despot who is moderately gracious as long as his unquestionable authority and power are acknowledged and deferred to; as soon as someone crosses him or even seemingly diminishes his might, he deals heavy, even deadly, blows against them. As I said, Falkland and Tyrrel are neighbors. Caleb resides with and has the confidence of Falkland. Barnes is Tyrrel's steward and manages his estates.
[Tyrrel] replied, "we will see what can be done. ... Hawkins, I think is your name? You may call upon Barnes, my steward, tomorrow, and he shall speak to you."
Thus Caleb and Barnes are neighbors. However, as Falkland and Tyrrel are rivals and enemies, at Tyrrel's doing, Caleb and Barnes don't really have any significant impact upon each other. In fact, Barnes' major role is in relation to the dramatic incidents centering around Hawkins and, later, Emily. Further, once Tyrrel is slain in Volume 1, there is little need left for Barnes at all.
Mr. Tyrrel was found by some of the company dead in the street, having been murdered at the distance of a few yards from the assembly house.