It is a matter of practice. I always teach people to try and hit every input possible. Also, this may help with how fast you are doing the directional input as that is the most important part to getting a special move out. Whenever you do the Cl.St.Hp try hold back at the same time. This takes away the process of pressing another input. the faster you can get the directional input out the more successful you will be. And for your ending input try to make sure your forward input is together with your mk input. Basically at the same time.
also your inputs should look like this (Back+Cl.St.Hp, Down Back, Down, Down Forward, Forward+Mk Axe Kick)