Sensor cleaning (actually AA filter cleaning) is a multi-step process....
1. Wet cleaning.... Is always needed if it's the first time a sensor has been manually cleaned, to remove oils that are bound to be on the AA filter, thrown off from the shutter mechanism. Any "dry" form of cleaning will only smear the oils, contaminating the tools and leaving something for more dust to adhere to. Usually a wet cleaning is a good start for subsequent cleanings, too. It's the best dislodging adhered particles.
I use the Copperhill method and tools mostly. That uses cheaper Pec Pads folded and taped in place on a cleaning wand. Pec Pads are essentially a rag-based paper. There are different size wands for different size sensors. You also can make your own wands and cover them with other products, but the Copperhill method is relatively inexpensive, the wands are a one time purchase, after all. Do not use common paper lens "tissues", they can leave fine scratches (paper made from wood pulp has minerals in it). Microfiber cloth can be used and is effective, but tends to be rather expensive. Only a brand new cloth should be used... cut down to size, folded and taped to the wand, and only used once.
Be careful to not get the swab too wet, either. Some of the ready made, pre-moistened swabs I've seen have been way to wet. A drop or two of sensor cleaning fluid is all that's needed. Shake off any excess before using. At best, too wet a swab will leave more haze on the filter... at worst moisture might seep in places you really don't want it.
2. After the wet cleaning there is nearly always some residue left behind, especially in the corners. I use a combination of things to remove that: Dust Aid low-tack adhesive pads (also used with a wand type tool), a small soft anti-static sensor brush, bulb (Rocket) blower, miniature vacuum, and a fine Speck Grabber tool for individual particles. It is essential that all oils be removed before using any of these, or you'll just contaminate the "dry" tool and/or smear around the oils, making matters worse.
A lighted magnifying loupe or eyeglasses can be a huge help doing this step. Just remember that things on the sensor are upside down and backwards from what you see in test images or in Live View, highly magnified (sensor dust cannot be seen in the viewfinder). So if you are seeing stuff in the upper lefthand corner of your images, on the sensor itself the stuff will actually be located in the lower righthand corner (assuming you are holding the camera with the lens mount facing you and in the horizontal/landscape orientation).
While you work sometimes it helps to set a standard household vacuum running nearby.... never directly sucking on the camera... just nearby so that there is some airflow away from the camera, while you work... Perhaps holding the camera face down so that particles can fall out and be drawn away by the vacuum. Just be careful of any small parts around the vacuum though (don't ask :rolleyes. It might be a good precaution to stretch a piece of old, nylon stocking across the mouth of the vacuum hose, held in place with a rubber band.
3. After all particles are removed, the final step is to lightly polish the sensor (actually the AA filter) with a Sensor Pen (same as a Lens Pen, but shaped to fit into corners better... and yes, supposedly this is what Canon does, too, as a final step after a wet cleaning). This is necessary because the wet cleaning usually leaves some haze. A Sensor Pen should not be used when there are oil and/or any particles still on the filter surface. Oil will be smeared and particles might scratch. This very gentle polishing leaves the surface pristine clean and helps prevent dust particles from adhering in the future, makes the camera's self-cleaning sensor cycle more effective.
Remember to have the camera's battery freshly charged and work reasonably quickly. The camera has to draw power continuously when in sensor cleaning mode, holding the mirror up and the shutter open. You don't want the camera to run out of power, close the shutter and drop the mirro onto tools you are using to clean the sensor. How long you have to work depends upon the battery type, age and condition, exactly how much time you have with a single, freshly charged battery. LP-E8s used in some cameras have half the capacity of LP-E6. The larger batteries used in 1D series have much greater capacity. Older BP511 vary a lot, but seem to have about 2/3 the capacity of LP-E6. To be safe, I'd try to finish in 10 or 15 minutes with most cameras (though a half hour or longer might be possible with many).
Not part of the cleaning process, but another silly little thing that helps.... Avoid using smaller apertures than necessary. This also helps minimize the effects of diffraction. With APS-C I try to keep to f11 and larger (with an 18MP APS-C camera, anything smaller than f7.1 starts to show some effect from diffraction). With a full frame, I try to limit myself to f16 and larger (the Diffraction Limited Aperture of a 21/22MP FF camera is f10). Regularly using f22 or f32 or smaller (depends on the lens) is rarely necessary for adequate depth of field and tends to shows up even the smallest speck of dust, plus loses some fine detail to diffaction.
Hope this helps!