I just submitted my PhD application, how should I write an email informing the Professor about my application? I emailed a Professor 2 months before the deadline. 
All I got was a neutral reply "thanks for contacting me and being interested in my research! It is great to hear that you are excited about that area.

Regarding your question of becoming a PhD candidate, I really appreciate you being interested in my research! In general I do NOT have open PhD positions at the moment. But you go through the admission process first. A committee makes the decision regarding admission. I hope this information is helpful"

Now that I have submitted my application, I feel that I should inform the Professor about this just to remind him about my profile/previous email. However, I am not sure what to mention in my email beyond the following 

"I just wanted to let you know that I have submitted my PhD application for Fall 2020 admission. I hope that the admission committee will select my application......."

Any ideas regarding what I can mention in this email.
It is typical in the US that a student chooses a dissertation research advisor only after admission. Often it comes after a few years of advanced coursework that prepares the student to take comprehensive exams (qualifiers). Up until then, you may have an academic advisor (for advice) but without the fixed expectation that they will guide your dissertation. 

So, you have hope that this person will find a slot after you arrive. You should do what you can to get close to the person, though, through their coursework and (especially) seminars. Maybe something will open up for you. 

But you can also keep your eyes and ears open for others who would be good to guide your work during this initial period. 

Note that students entering with an MS and having already done sufficient coursework that they can take comps early have a shortened time scale, but probably still an opportunity after you arrive. It isn't essential, here, to have the final advisor on board at admission. 

Other places little of the above applies and one often enters the specific research group at admission. 



If you want to write the professor, pretty much anything will do if it just expresses your hope that you get a chance to work together in future. The emails you have received are entirely typical it the US.