When composing letters, I often use this as an opening statement (not including the normal Cher ... etc.)
Je vous présente mes salutations chaleureuses.
Does this sound appropriate in French, or does it appear to Anglicized?
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While this does not sound anglicized at all, I do not think it is something someone would say as an opening statement.
Indeed, if we usually use salut as an opening statement in an informal conversation, a salutation in a letter definitely sounds to us as a closing statement.
In French, opening statements in letters tend to be really simple and concise : for example most letters simply start with a
Madame, Monsieur in a formal letter or Cher / chère in an informal letter
French correspondence usually uses such formulations as the one you wish to use at the end of the letter, where you can also find, for example :
Je vous prie d'agréer mes salutations distinguées
(You can find more of these in the answers of this question : Quelle formule de politesse doit-on utiliser à la fin des documents ? — Which valedictions can be used to end a letter?)
However, if you feel like you MUST use Je vous présente mes salutations chaleureuses at the beginning of your letter (which I would most likely not recommend), please note that in this context, it would be better to swap chaleureuses with salutations, it will sound more professional :
Je vous présente mes chaleureuses salutations
If you are looking for other formes de politesse, as we call it in French, the link Laure provided is really useful and I used the forms provided there more than once when I was not sure about how to finish a letter ! Here it is : http://www.lettres-gratuites.com/formules-politesse.html
Hope I could help !
As Laure said, salutations are generally put at the end of the letter. Her link However, you can ask how the person is doing, or wish her good, like that for instance :
J'espère que tu vas bien, etc.
Comment vas-tu ?
But your sentence is too close to the kind of salutation you put at the end of the letter.