(Please, for this question I need the assistance of a person whose
mother language is English)
How do you reply politely to a letter/email ending like this? Suppose
that this is a formal email, related for instance to job searching,
from the company where you have applied for:

"...and of course I will keep you posted.
Kind regards
(Signature)"

Is it okay whether in my brief reply I use:

"I look forward to hearing from you" 

Since the writer in the first letter wrote "OF COURSE, I will keep you
posted", would it be a bit ANXIOUS or demanding to end the reply as
stated: "I look forward to hearing from you." Or, on the contrary,
both are common and "automatic" expressions widely used? Please
ellaborate.
Kind regards,
Tom

Hi tom136,

"I look forward to hearing from you soon" is a great response to
"...and of course I will keep you posted." Your potential employer has
chosen to end his letter in a positive way and therefore you should
respond in kind. Honestly, I cannot think of a better response - in my
opinion "you nailed it".

Both phrases are commonly used, but I think in this case, the employer
has gone a bit further than an automatic response and has ended with a
positive "action" phrase - it is appropriate that you reply with a
positive "action" phrase as well, which you have. You will not sound
anxious or demanding - if you did not respond with an appropriate
promise of action, you would run the risk of sounding nonchalant or
uninterested, which would not be a good thing to do.

This, as I said, is my personal opinion but I have taken the time to
do a bit of research online to make sure that I'm not "out to lunch"
on this and also I've asked a colleague their opinion and I can say
with confidence, "not to worry, all is well".

That all said, if you are concerned that you do not pick up on all of
the nuances of the English language, it would be entirely ok to
mention the fact at the end of your letter (assuming the employer
already knows that English is your second language).

Here are a few letter writing links that I hope you find helpful.

1. Businessletterpunch.com: step-by-step guide: 
"This web site takes users through the actual steps of writing a
business letter."
http://www.businessletterpunch.com/

2. Business Letter Writing Basics
http://esl.about.com/library/weekly/aa041399.htm

2. Essential Resources for Writing Business and Employment Letters
http://jobsearchtech.about.com/library/bl-business-letters.htm

4. Writing an Effective Business Letter.
http://www.business-letters.com/business-letters.htm

I hope this helps. If you have any questions, please do not hesitate
to post a clarification request before rating my answer - I will be
happy to respond.

Thank you and Good Luck!
hummer

Terms Used:
"business letters"
"business letter writing"

Dear hummer-ga:
Thank you for your precise reply.
Kind regards,
Tom

FWIW, I agree.

Your phrase does not so much indicate anxiety, as keen, on-going interest.

Dear xarqui-ga:
Thank you for your comment!
Tom

Dear Tom,

Thank you for your thank you and fine rating, I'm happy you are happy.

Take care,
hummer

"I look forward to hearing from you soon" is, IMHO, an excellent and
courteous way to show that you are receptive to the employer. In no
way does it seem overly "anxious". There's a difference between being
avidly interested and being neurotically anxious.

Now, if you were to close a letter by saying "Pick me, pick me,
pleasepleasepleaseplease pick me," I would say that you were going a
bit over the edge. ;)

Magnesium:

Tee hee hee :-)