The speaker in your two examples is British, with the characteristic "EE" sound of that accent. The way he pronounces both "feel" and "feeling" is greatly exaggerated, in order to allow you to hear each part clearly. You should not try to imitate this way in natural speech.
I can't much help you with that accent, but I can say that a "standard" American accent will certainly pronounce the final "L", although not as if it was a second syllable. FEE-ul would be too much. End the word with your tongue pressed to the roof of your mouth, and that should be sufficient.
With "feel", You pronounce the "EE" sound fine. Your challenge is to learn to pronounce the "L" sound as distinct from the "R" sound. Once you get that, the word will sound more natural. Your second recording is better, so just learn to make that same sound more quickly, with less exaggeration.
Additionally, many English speakers will blend the final "L" of a word like "feel" with any following word that starts with "L":
I fee'like going to the park today.
Does she fee'lonely?
And so on.
"Feeling" is two syllables, each of which is pronounced. The "L" sound between the syllables can be very brief, with only a quick touch of the tongue to the roof of the mouth.