@Ellis2010 To begin with, it is totally fine to only perform the exercise on strings 4, 3, 2 & 1. The reaching over and stretching out does become more difficult for beginner hands on the 6th and 5th strings.
In time, you should find you have greater flexibility and reach and can step across to string 5, then string 6 eventually. 
You want clean and clear notes in your playing, of course. In the beginning stages of this exercise, the main aim is to improve finger dexterity and if you lose a little of the musical clarity in your notes as you first get your fingers working then do not worry about it all that much. It should improve as your ability and skills from practicing the exercise improve also.
ps
Have you figured out recording and uploading a video yet?
That is fine. You are not playing chords where multiple strings need to ring out cleanly all together. You are playing a single string exercise. Muting unwanted noise from adjacent strings is totally okay.
@Bor_MS Michael. Kudos for sharing a photograph. I wish I was there in the room with you to help with your posture and guitar position. Please try to adjust as you have already found it is leading to pain.
Angle the guitar neck up, not down. That will automatically lift your elbow off your left leg and give it freedom of movement and be an essential part of the fretting process unhindered. Sit up straight and try to refrain from leaning over forwards to stare at your fretting fingers with a curved back. You will see those fretting fingers more easily with the neck angled upwards.
Here is Justin’s position in this exercise.
