Here's some tips on giving a good backrub to someone in a chair, which is probably the most common way do do it. We're going to call the person getting the backrub the patient, because we need to call them something and it's denotationally accurate even if the masseuse/masseur is not a doctor. Have the patient sit in an armless chair, reversed, with their arms resting on top of the chairback and their chin on their arms. As far as possible, you want to allow the muscles you're working on to relax; relieve them of their job of supporting whatever body part they support. This chair position allows a pretty decent amount of relaxation in the shoulders and lower back, usually the worst problem areas. The middle back will be somewhat tensed with keeping it straight, but without a table you can't have everything. Never fight the muscle: never try to push it out of place when it's tensed to provide actual support. That hurts and does nothing for them. If the patient wears a bra, it's best if they remove that while leaving the shirt on, or failing that it's good if they just unsnap the back - tight straps really get in the way.
Start off with what they expect; kneading their shoulders with your hands. Don't keep that up too long, though, because your hands will wear out quickly. Move to working the large back muscles on both sides of the spine with your knuckles or the heels of your hands. It's the same sort of thing you started out naturally doing with your thumbs, but now you can use your body weight by leaning in rather than just hand strength. If you're small, you'll want to do a lot of your work from standing up, if you're big, you'll mostly want to sit behind them. Without oil, when you are working deep use very short strokes and frequent repositioning.
Work your way down on both sides of the spine; use your knuckles especially near the shoulders and on problem spots, and the heels of your hands after you get down past the ribcage. Feel where the thick muscle is and where is ribs or thin muscle over organs; go very lightly except on thick muscle. When you get down to the pelvis, if their clothing allows it (belts get in the way), work out to both sides then back in again. Work up and down several times. Work the rhomboids between the shoulder blades.
The muscles stretching from the shoulders to the neck and the neck itself usually really need work but are a problem from a sitting position. The best solution I've found is to have the patient lean back against you with their head on one of your shoulders and use the hand strength you've been saving, and just knead the muscles. If you can have the patient lie on their back, on a bed or on the floor, it will work much better; have them do so, then use your knuckles or the edge of your hands to work the upper shoulders. Rub the muscles in the neck much more lightly, they're comparatively small. The sides of your thumbs are good for that if they're leaning against you, or the sides of your fingers if they're lying down.
After your hands are pretty much tired of deep work, try an effleurage stroke. With them in the chair leaning on the chairback again, or flipped on their face on a bed, put your hands together flat against their back at the base of the spine. If their clothing will permit, stroke with a flat hand and moderate pressure up to the neck, bringing your fingertips with light pressure to the base of the skull, then out over the tops of their shoulders, back in toward the base of the neck, hands back together, and back down to the base of the spine. At the base of the spine, stroke out to both sides of the pelvis, then back in to where you started, and repeat. The trick with this stroke is smoothly modifying the pressure depending on where you are and where they're sore. Rook, for example, is very sensitive on her lower back, so the pressure must be very light there, but most people like moderate pressure there. Too much pressure on the neck is bad, but a lot between the shoulder blades (on the rhomboids) is usually good.
There's something that may be some kind of special natural talent that some people have and some don't, but it's always seemed to me that everybody could do this if they would focus their mind and get their heart into it. That is feeling what would feel good to the person you're working on. After I get started and into it, I can kind of sense what kind of pressure (heavy, light, broad, pointed) is needed in which place. Open your mind and imagine what would feel good, and then try what you imagined.