Convergence refers to the control of the instantaneous positions of the red, green, and blue spots as they scan across the face of the CRT so that they are as nearly coincident as possible. Symptoms of poor convergence are colored borders on solid objects or visible separate R, G, and B images of fine lines or images,
Note: It is probably best to face the set East-West (front-to-back) when performing any purity and convergence adjustments. Since you probably do not know what orientation will eventually be used, this is the best compromise as the earth's magnetic field will be aligned mostly across the CRT. This will minimize the possible rotation of the picture when the unit is moved to its final position but there may be a position shift. Neither of these is that significant so it probably doesn't really matter that much unless you are super fussy. Of course, if you know the final orientation of the TV in your entertainment center - and you don't expect to be redecorating, use that instead. Or, plan to do the final tilt and position adjustments after the set is in position - but this will probably require access to the inside!
First, make sure no sources of strong magnetic fields are in the vicinity of the TV - loudspeakers, refrigerator magnets, MRI scanners, etc. A nearby lightning strike or EMP from a nuclear explosion can also affect purity.
Cycle power a couple of times to degauss the CRT (1 minute on, 20 minutes off) - see the section: Degaussing (demagnetizing) a CRT. If the built in degaussing circuits have no effect, use an external manual degaussing coil.
Assuming this doesn't help, you will need to set the internal purity and/or convergence adjustments on the CRT. Modern CRTs usually use a combination of a series of magnetized moveable rings on the neck, and yoke position and orientation to set purity and convergence.
First, mark the positions of all adjustments - use white paint, 'White out', or a Magic Marker on the ring magnets on the neck of the CRT, the position and tilt of the deflection yoke, and any other controls that you may touch deliberately or by accident.
However, if your set is still of the type with a drawer or panel of knobs for these adjustments, don't even think about doing anything without a service manual and follow it to the letter unless the functions of all the knobs is clearly marked (some manufacturers actually do a pretty good job of this).
Note: some CRTs do not have any adjustable rings for purity (and static convergence). Either an internal structure in the neck of the CRT or an external 'permalloy' sleeve is permanently magnetized at the factory and there is not way of tweaking it in the field. However, it may be possible to use a normal set of magnet rings in addition to or in place of it to correct for purity or convergence problems due to loss of magnetism due to age or someone waving a 10 pound magnet near the CRT neck!