You can do the select part with a controller in place of the mouse, but the actual copy and paste commands I think are only available on keyboard, at least so far as I've seen. (Most PS3/PS4 players, including myself, play primarily with controller but also keep a keyboard plugged into the console for chat. The same CTRL+C and CTRL+V commands you'd use from a PC are also available from a PS4 console's keyboard.)

For the select part, you can use the PS4 controller's touch pad as a mouse trackpad, or you can use L1+R3 to go into "mouse mode", where the analog stick moves a mouse cursor and L2 and R2 act as left and right mouse buttons. (Note, the L1+R3 to toggle mouse mode on or off doesn't work while in a window that assigns a different meaning to L1, such as the chat window that uses it for changing tabs. You have to unselect that window before turning on mouse mode.)