I do a 4-5 mile walk daily. When I stop, I feel a little dizzy and have slight fluttering in my chest. Is this serious?

May or may not be... Slight fluttering could be normal depending on how fit you are, how fast/vigorously you are walking, and how long you have been walking in such a way. People normally feel increased heart rate associated with some palpitations after moderate physical activity. Dizziness is not necessarily normal. It would be best if you go see your doctor to be evaluated.
Urgent MD evaluation. You are apparently describing a potentially significant cardiac (heart-related) condition tha will require a thorough evaluation by your healthcare provider. Seek medical evaluation now.
Usually not. This may be nothing - fluctuations in BP for example after exercising. Just being able to walk 4-5 miles a day lowers your overall risk of having a bad heart problem. However talk to your doctor about two options. One, have a stress test performed to see how your heart does when stressed. Two, consider a holter monitor to see exactly what your heart is doing when you feel the symptoms.