OK, I'm using a Fender FT-100 chromatic tuner to tune each string. Everything goes fine and I think I have it in tune. Then when I play the 5th fret on the E string, it's an A that is little sharp and doesn't match when I play the A string. If I keep going down the 5th frets, they are all a little sharp when compared to the open strings.

Is there something I can do to correct this or should I just tune the E string and then use the 5ths (and the one 4th) to tune the rest of the strings?

Its going to be a little bit sharp because your fretting it, thats why i dont like tuners. Ive always done it by ear and harmonics. Just do whatever it takes to make it sound right.
its most likely your intonation. look up about that. also ur not going to get it perfectly in tune just real close but yeah if its off by a lot then yeah check your intonation
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What the other guys say is true, but it might also be that your neck isnt straight, or your bridge isnt correctly adjusted. For that, I suggest you go to a guitar shop to get it right
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Ok guys, thanks for the help. Makes perfect sense now! I was fretting it a little hard and made it sharp. So I just tried it again and fretted lightly. Much better.
I also checked the intonation and it is off slightly on a couple strings. I can fix that.