What is the first order logic statement of "this field is of characteristic zero"? I want to state that a field $F$ is of characteristic zero in logical notation to an audience without referring them to the meaning of the characteristic of a field.
My first thought was the proposition
$\forall x \in F \setminus \{-1\} : x + 1 \ne 0,$
but this holds for any field.
How can “of characteristic zero” be stated using quantifiers?
 A: No.
The statement is equivalent to

for all $x\in F$, if $x+1=0$ then $x=-1$.

This is true in every field.
To explain characteristic zero, how about just "adding $1$ and $1$ and $1$ and so on [finitely many times] will never give you zero".  Omit the "finitely many times" if you think it will confuse your audience.  And I would certainly suggest briefly giving an example of a field with non-zero characteristic, otherwise they will just say "that's obvious, what's the point?"
A: No. In any field the equation 
$$
x + 1 = 0
$$
has the unique solution $x = -1$, independent of the characteristic.
If your audience is unfamiliar with the characteristic of a field and you need to explain it, do that with an example. In $\mathbb{Z}_5$ you have
$1 + 1 + 1 +1 + 1 = 0$. In a field with characteristic $0$ no sum of $1$s can be $0$.
Don't use logical symbols, use words.
Edit in response to the edited question.
If you must use a formal statement you might say

For every positive integer $n$ the sum of $n$ $1$s is not $0$.

or

The smallest subfield is infinite.

