First, install a brand new needle. Verify that it is the correct needle and needle eye size for the thread and fabric you are sewing. (Change needles frequently. Needles are the most inexpensive sewing accessory and most the most common repair for a malfunctioning sewing machine.)
What you are experiencing sounds like the thread is shredding at the needle eye and eventually breaking. Sometimes, a top stitch needle will work better because the groove in the back of the needle shields the thread better as the needle pierces the fabric multiple times. A piece of thread can pierce the fabric over a dozen times before it forms a stitch, so friction and rough fabric will take its toll.
All About Needles
Use only good quality threads--AVOID old thread and bargain bin threads. They are cheap and there is a reason why they are--because what they make up for in price is paid in frustration!!!! Polyester thread used to be the bane of sewing machines, but polyester thread has greatly improved and is now a favorite for experienced sewists. It is thin and strong. Threads that are fuzzy (cotton-wrapped polyester of old Coats & Clark variety) threads are not the best for your machine or project. Check out better quality threads, ie Gutermann, Superior, Sulky, Aurfil, Isacord, and multiple other brands. Make sure you are buying the correct type of thread for your project--don't use embroidery thread to sew your garment unless you want the seams to pop open while you're wearing it.
Remove the thread from your machine. ALWAYS RAISE the presser foot and then rethread from the beginning. The raised presser foot releases the tension disk so the thread will seat properly.
Check that the upper tension is set somewhere around the midway point. This is a factory standard although most machines may require some adjustments. Test the stitch on your fabric. The tension is ideal when both upper and bobbin threads meet in the middle of the fabric. If the bobbin thread shows on top of the fabric, the upper tension is too tight. If the upper thread shows on the underside of the fabric, the upper tension is too loose.
If the thread is still shredding, then there's a likelihood there is a burr or nick somewhere on the needleplate or in the bobbin/hook area. If you wrap your index finger with fine fabric (like old hosiery) and rub you finger around, you may find the burr. If it is small, you can lightly smooth it with a fine emery board. If it is a large one, you may need to replace the part.