1990 BMW 318i central locking problem

As for vehicle made in 1990 and when it comes to BMW I'd suggest you to contact garage, to solve BMW's problems, but in other hand it sounds like complete electronics malfunction, so it might be expensive, asspecialy when it comes to 19 years old vehicle. It might take a while to solve it, but we can try.
So before we start stripping your car I need to know couple of things.
Dooes your vehicle has an alarm system controlled by the same remote, or is it for imobiliser and/or central locking only?
Would be nice if you would have any ideas about the type of alarm and/or imobiliser system, as it would be easier to get to solution.
Try to lock yourself inside the car and when locking listen very carefully where does the clicking sound comes from, as for lights relays are being used, so by finding the source of clicking sound you'll get to the location of alarm/imobiliser/central locking system. So after that we will know where all the work is needed. Also I'd like you to note any identification marks from the unit. Usually it is being placed under the panel, next to the passengers right feet but in some cases it might be just above the dirvers knees, under the dashboard, rarely even under the back seat. so first thing that we need to do is to find the exact unit.

Also as for so old vehicle try to get to the wires which goes from the car body to the door (passenger), as due to age wires might be snapped or damaged, preventing doors from lock/unlock commands.
If unable to lock manually you nead to strip out the door-lock itself, wash it with something like WD-40 and add some grease. Nothing electronicaly can prevent manual operation of the door-lock. not in 1990.

So if you are not experienced try to use this message backwards, and I bet you'll get the results before the message ends.