I know this sounds really generic/obvious but you need to do an overload on pushups, do as many as you can do physically, it should be so much you can barely move.

That's how I got started and how I got from being barely able to do barely a set 10 pushups to being able to do a set of 80 fairly easily.

I used to help my dad lift hay in the summer (130 acres of hay filled fields, you do the math), and after lifting all the hay, I was exhausted, but I had some serious upper body strength. If pushups are too difficult, do some good old fashioned manual labor. But it finally occured to me lifting hay put me in shape when I could do as many pushups as the really athletic kids.