How long on a background check does a DUI charge and a reckless driving conviction take to come off?

In KY, there is a 5 year "look-back period".  This is the period of time in which the court will look-back into your driving record for sentencing purposes.  Thus while a conviction will stay on your driving record forever, if you have not had an offense during this 5 year period, you will be considered to be a first-time offender.  So your reckless will not effect you if there are no further offenses within this time period.  And in your case, the look-back period doesn't even apply to your DUI since you weren't convicted in the first place (this is why your MVR is clear). 

However, that is your driving record.  Your criminal record is something different.  In order to clear that you must "expunge" your record, otherwise arrests/convictions will remain indefinitely.  To do this, you should speak with an attorney in your area that handles expungements, they can best advise you as to your state law.