It is much easier to make a doll with a sewing machine but it's totally possible to do it by hand. I did 3 on a vacation one time all by hand. Just make sure your stitching is well tied and knotted down and is not spaced too far apart. Basically, make sure it's secure so no stuffing can get out. Look at reference pictures of the character you want to make. Write down what type of fabric you need for them and what colors. Make sure to bring this list and the picture (for correct color reference) to the fabric store with you. I make all my dolls out of fleece (NOT felt). You probably will too unless you need a specific feel like leather (pleather), something silky, or cotton. On a more complicated note, you may have a doll that needs some part of it to stand up. The Midna doll I made needed this for her headdress. Use the stiffest interfacing you can find at your fabric store (excluding the plastic ones). These are waterproof and work great in between two pieces of fleece. Don't worry about ironing it, just sew it in between. Also cut it smaller than your fleece pieces. You don't want it to show.
As for making eyes, which is one of the hardest parts, just do it like drawing. Cut out the white of the eye, the black rim (keep it stable! Don't cut something that's falling to pieces, if you must use felt because it's stiffer), the color of the eye, pupil, and white shine. Put them together in that order. Sew the pupil onto the color of the eye, then the white shine. Sew the white of the eye onto the doll's face and then the black rims. Finally position the eyes and sew the rest on. Don't forget eyebrows and a nose! (also, do ALL face, ect. details you can and sew them onto the front/back pieces of the doll before sewing the whole doll together.
For ears, pin them on backwards: ears on edge of doll's head with them pointing in towards the face. Once you stitch the pieces good-side-to-good-side and turn it inside out they will stick out like ears do.) When you're ready stitch the front and back pieces together good-side-to-good-side and turn inside out! Stuff your doll with stuffing (duh) and hand stitch the hole up! Hug at least three times upon completion! If you have more questions ask me!