How Do You Clean The Ice/water Dispenser Line In Fridge?

$50 is quite a bit to pay for a filter. You might want to consider a pre-filter (a filter in front of your $50 filter), which can be bought for around $10-15.
Run water through your new filter for several minutes before connecting the supply line to the refrigerator. This will flush out any loose carbon particles. Many sources recommend using only a copper supply line. I personally have what I think is a high density polyethylene (plastic) supply line and have not had any problems.
You might try disconnecting the supply line at the point it enters the refrigerator and tasting the water in the line. This will help isolate the source of the problem. If the water tastes fine, then determine whether the ice has the same weird taste as the drinking water. If not, I’d suspect a bacterial growth in the water tank. Look in the homeowner’s guide for your refrigerator to see if there are instructions for cleaning the water supply. If not, I suggest using a relatively harmless solution of hydrogen peroxide.
I’d like to add that ice, while it is freezing, will pick up a taste from odors in the refrigerator. Always wrap your onions and garlic.

Just change out the line.

Do you have a copper water line? How about the rest of the pipes in the house? You say you have a softner, they use salt to soften your water, maybe your filter is being affected by a combination of all these factors. Try bypassing the softner, and see if that makes a difference, or try changing the line to the fridge. Good luck to you!

I would change the tubing that came from the supply and yes spend the extra 50 on a new filter..I have this too and the minerals in the water tend to stop it up so its good to change the line and filter every so often to assure you have excellent clean water.Really its worth the money..