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It's a fairly simple process provided you can work safely on the roof at the chimney. You buy a stainless steel liner of the appropriate length and diameter. It comes with a cone you attach to one end. You secure a nylon rope to the cone then drop the line (with a weight on the end) down the chimney. Your guy in the basement then pulls on the rope while your guy(s) on the roof feed the liner down the chimney. Then you mortar up the chimney are the liner in the basement and install the top/termination kit at the top of the chimney.
If tenants away from the boiler room but near the chimney are complaining of the smell of smoke/diesel then a chimney problem is a pretty safe bet. There are no "tests" except perhaps a pressure smoke test but it's unlikely to tell you anything that you don't know already.