If a suitable unicode char exists for a combination \not + unicode symbol you can add the needed declarations and they will work like the combination of \not + command. But imho there is no \not\approxeq, so here one needs a fallback like a combination with a solidus. You get a 6 in this case as there seem to be a bug in unicode-math: it looses the definition of \not from unicode-math-tables. This seems to be a work around (but imho doesn't looks perfect).
\documentclass{article}
\usepackage{fontspec}
\usepackage{amsmath,amssymb}
\usepackage{unicode-math}
\setmathfont{Latin Modern Math}
\ExplSyntaxOn
\cs_gset:cpn { not≤ } { \nleq }
\cs_gset:cpn { not≲ } { \not\lesssim }
%Fallback bug correction:
__um_sym:nnn {"00338}{__um_oldnot: }{\mathaccent}
\ExplSyntaxOff
\begin{document}
With \LaTeX{} commands:
\begin{gather}
A \not= B\
A \not\le B\
A \not\lesssim B\
A \not\approxeq B
\end{gather}
With Unicode characters:
\begin{gather}
A \not= B\
A \not≤ B\
A \not≲ B\
A \not≊ B
\end{gather}
\end{document}

Edit
One get a better "fallback" \not if one uses the overlay keyword. This works only with lualatex: As an example here for for your \approxeq:
\documentclass{article}
\usepackage{fontspec}
\usepackage{amsmath,amssymb}
\usepackage{unicode-math}
%\setmathfont{Cambria Math}
\newcommand\napproxeq{\Umathaccent overlay fixed 7\symoperators "00338 \approxeq}
\begin{document}
$A \napproxeq B $
$A \not\approxeq C $
\end{document}

I will add this to the issue tracker for unicode math.
\notis very broken inunicode-math. – egreg Aug 11 '16 at 22:02\setmathfont{XITS Math}or similar doesn't change anything. – Clément Aug 11 '16 at 22:02\not\approxeq– Clément Aug 11 '16 at 22:03\notapproxeq:\def\notapproxeq{\mathrel/\joinrel\mkern-8mu\approxeq}and use it like$A \notapproxeq B$– alwaysask Aug 11 '16 at 22:21unicode-math, for just one symbol? – Clément Aug 11 '16 at 23:58\def\notapproxeq{\,\mathrel/\joinrel\mkern-8.5mu\approxeq}. It turns out the\notdefinition in unicode-math looks for an already defined \not version of the symbol so just by inserting the new definition in the preamble theA \not\approxeq Bbit gets fixed. – alwaysask Aug 12 '16 at 07:04≠(\ne),≰(\nleq),≴(\nlesssim)? Unfortunately,≊̸is not mapped byunicode-math, so you have to build this one yourself:\def\napproxeq{\mathpalette\donot\approxeq} \def\donot#1#2{{\ooalign{$#1#2$\cr\hidewidth$#1\divslash$\hidewidth\cr}}}or use≇(\ncong). – Henri Menke Aug 12 '16 at 15:35unicode-math.sty, January 8, 2018. – sgmoye Jan 09 '18 at 18:36