Good morning! The my morning 

This morning at fifteen past six, I went by bike to make a run... I was able to do five kilometers..
forgive me for the mistakes ... but I do not study English from high school...
a little patience.. 
Bye :)

Good morning! My begin day...You can say " My day begins....or I bagan my day...."

This morning at fifteen past six, we typically phrase this by saying six fifteen.  the way you used it is correct but other options and more common potions are six fifteen.....or to say a quarter past six. I go to the bicycle path to make a run..try:  I ran on the bicycle path. . I was able to do five kilometers..
forgive me for the mistakes ... but I do not study English from middle school..
a little patience,,,
Bye :)

no patience needed you di a great job....just offering options that have a better flow. 

Good morning! This is how my day began.

This morning at quarter past six (or six fifteen, but most Native English speakers will say 'quarter past'), I went for a run on the bicycle path (also called bicycle lane). I was able to do five kilometers.

Forgive me for the mistakes..but I did not study English in middle school (or 'I have not studied English since middle school')

a little patience,,,

Bye :)

(well done!)