How can I hang a picture without putting it in a frame?

It depends on what the picture is, that is, what surface it is placed on. An oil painting that is on a "frame" of wood, can be hung by putting a wire on the back and hung like that. For what essentially is a piece of paper (a print, a photograph) you could put up a bulletin board and fasten to that or you could just glue it to the wall (would be removal like wall paper is but you might ruin image in process), you could use a piece of mat board, fasten picture to that, cut to dimensions you like glue some string on the back of the mat board and hang like that. Lots of ways to achieve what you want.

Roll up tape on back and stick to wall.

I'm confused 😕 Maybe I haven't had enough coffee! Do you want to make a frame or hang the picture without a frame?

Command adhesive to make strips for wall hanging lighter weight items.

Poster Putty can enable you to put frameless photos on any surface.