How To Get Flowey Back?

Go to Users\yourusername\AppData\Local\ There should be a folder called "UNDERTALE". If you want to remove absolutely everything, delete that folder.

If you're playing on the Steam version, make sure you disable Steam Cloud for the game as well.

He's still there, but you do miss some of his dialogue, as indicated in another answer. A True Reset isn't necessary to get him back, almost any reset will do - if Flowey survived your last run, he'll be at the beginning of your next one. If you killed him last time, of course, he won't confront you at the beginning of the game, because

he remembers your previous runs unless you do a True Reset, and is afraid to confront you because he can't Load his Savefile while you're around - and you freaking killed him!

Manipulating the save files is only necessary if you absolutely don't want to spend the time to complete the game again to get him back AND absolutely want to hear his dialogue early in the game. In Genocide, it might be worth it, as before you leave the Ruins he has one of only three voice-acted lines in the game if you qualify for Genocide when you go to leave.

You can still get any ending without Flowey being present, though you will miss any dialog he would have before the final encounter with him.

A True Reset will be sufficient to restore Flowey's presence in future playthroughs. However, erasing the save data (as mentioned in the other answer) would get the same results faster.