The following drawings show both a ray-trace of the telescope plus a view of what you would see while looking through the Cheshire eyepiece. While the Cheshire views are drawn fairly accurate, the ray-trace and optics are shown grossly over tilted and miss-positioned so that you can see what should be happening to your telescope as you make adjustments. Also note that different systems are keyed to different colors. Green represents the Cheshire eyepiece in both the eyepiece view and the ray-trace. Purple represents the secondary, blue the primary, white the paper reinforcement ring, and gray reflections on the primary. The small gray doughnut inside the secondary reflection is the reflection of the underside of the Cheshire. Yellow dashed lines show the path of a star's light entering the telescope and eventually reaching the eyepiece. In the ray-trace the site lines are shown dashed and are also keyed by color. Of special importance is the green site line of the Cheshire cross-hair which is shown as it reflects through the system. The purple dashed lines represent the outline of the secondary. Blue dash lines are the edge of the primary (not necessarily where it is reflecting an image). Also shown in green are the site lines representing the field of view of the Cheshire when one is looking through it.
Collimation always begins with the eyepiece and ends at the primary. Because of this the first step is to make the focuser as square as possible with the tube. A premium focuser can be adjusted by turning allen screws that rest against the main tube. In this way the tilt can be easily changed. Some focusers will have to be shimmed to accomplish this goal. One method that works well as a first attempt is to remove the secondary and mark the tube directly opposite the focuser bore. Then with the Cheshire installed the tilt can be adjusted until the cross hair is over the mark. However the true test will be how the secondary looks when viewed through the Cheshire. The outline of the secondary should appeared centered in the Cheshire. As you can see below our first attempt to square up the focuser has the secondary centered from side to side, but front to back it is a bit too far toward the primary. By the way this procedure assumes that the secondary's spider has been centered in the tube, so that from side to side the secondary is also centered in the tube. The spider's center hole should be positioned slightly away from the focuser to allow "offset". This amount of offset is much debated but can be calculated using various formulas. If your system is faster than f/6 this should be considered. If the system is f/6 or slower the secondary can simply be centered both ways. When I build a Newtonian from scratch I try to drill the mounting holes for the spider so that the center has the correct offset without distorting the spider. At this point we are ignoring the images visible inside the secondary (gray, white and blue).
Second, the tilt of the secondary should be corrected. Even though our first guess of focuser tilt and secondary position may not be quite right, the tilt of the secondary needs to be set so that we can further refine the secondary's position. In the drawing above the secondary is close to the correct position, but where it is "looking" is wrong. The paper ring (white) is not under the green Cheshire cross hairs. This tilt can usually be adjusted by turning three push-pull screws on the rear of the secondary assembly. On most telescopes this will also cause the secondary housing to move slightly from side to side as well so it may be necessary to re-adjust the side-to side position (or the tilt of the focuser) to keep it centered. As the three screws are turned the paper ring will appear to move in relation to the cross hairs (at this point ignore the Cheshire reflection shown in gray as well as the primary outline shown in blue). Keep adjusting until the ring appears directly under the cross hairs as shown below. Note that the purple secondary is still not centered inside the Cheshire, but that the paper ring is centered. The secondary is now tilted correctly, but its position is wrong and precious light would be lost if this were not corrected. In the diagram below the secondary must be moved forward in the tube as well as toward the focuser.
Third, the spider should be adjusted or the tilt of the focuser checked until the secondary is centered both ways in the Cheshire. Then adjust the tilt again. This may cause the secondary to become un-centered once more, so repeat the procedure until the secondary is both centered and the paper ring is on the cross hair. The drawing below shows this. Also it is important to note that the outline of the (blue) primary now appears centered in the (purple) secondary. If it does not then the secondary is still not quite in the correct position. Also, remember not to cheat and look through the focuser without the Cheshire as you will unconsciously move your eye to make the primary appear centered. Finally make sure that all of the primary is visible inside the secondary. Remember to rack in the focuser untill your eye is close to the focal plane. If the primary clips and slightly beyond the edge of the mirror are not visible, then the secondary may be too small! Once the primary is centered in the secondary and the secondary is centered in the Cheshire and the paper ring is at the cross hairs you are done with the secondary adjustments. This to me is the hardest part of the whole process. We are still ignoring the reflection of the secondary and the Cheshire (gray) inside the primary.
Fourth, it is now time to adjust the tilt of the primary. This tilt is detected by observing the location of the Cheshire reflection (gray) that we have been ignoring up until now. As you may have noticed the Cheshire has a notch cut out of its side that allows light to illuminate a shiny surface inside the Cheshire. This shiny surface is what we see as a reflection in the primary. By turning the screws on the back of the primary's mirror cell the reflection will appear to move in relation to the fixed cross hairs and the paper ring (white). Some cells are spring loaded and this process is fairly easy. Simply turn one of the adjustment screws until the reflection comes as close as possible to the paper ring, then proceed to a second screw. Turn the second screw until the reflection again comes as close as possible, then to the third screw. Turning the last screw should bring the Cheshire reflection on top of the paper ring. If the reflection is still not quite centered then repeat the steps above by re-adjusting the three primary screws. Some mirror cells use a push-pull arrangement and have six screws on the back of the cell. One screw loosens, the other tightens. These are more difficult, but with a little practice the Cheshire reflection can be placed in its correct position. If all this is done correctly the view should look as shown below. The secondary outline is centered in the Cheshire. The primary is centered inside the secondary. The Cheshire reflection and the paper ring are both under the Cheshire's cross hairs. At this point the telescope should give very good images. Stars at the center of the eyepiece field should appear round and as they are defocused they stay concentric with their focused position. You have collimated a Newtonian!
As a final note observe that the secondary's reflection played no part in this process. Some people think that it should and this is a mistake. If the telescope is properly collimated the reflection of the secondary will not be centered with respect to the Cheshire and paper ring. Some dashed gray rays have been drawn on the ray-trace to show why this occurs. Also the above drawing shows why an offset is required for the secondary. Since the side of the secondary furthest from the eyepiece is closer to the primary than the side closest to the eyepiece, it intercepts the cone of light from the primary earlier when it is wider than when it intercepts the near side. If the system is faster than f/6 some of the light would be lost that comes from the far side of the primary. Images will still be good, but not as bright. The main objectives in order to have good images are to have the primary perpendicular and centered on the optical axis and the eyepiece centered and parallel to the optical axis as it is reflected from the primary by the secondary.