First of all, \biggl. and \biggr. do exactly nothing, except introduce unwanted opening and closing atoms (and, possibly, the relative spacing).
Second, \bigl/ makes the bigger / an opening atom, so you should use \big, instead.
Third, the bigger / has wider sidebearings than the normal size symbol. If you need it only at normal and \big size, here's a way:
\documentclass{article}
\usepackage{xparse}
\NewDocumentCommand{\dslash}{s}{%
\IfBooleanTF{#1}
{\big/\mkern-7mu\big/}
{/\mkern-6mu/}%
}
\begin{document}
$X\dslash H$
$X\dslash* H$
\end{document}

If you want other sizes, then something more complex should be devised. Here's an example:
\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand{\dslash}{O{}}
{
\str_case:nn { #1 }
{
{}{/\mkern-6mu/}
{\big}{\big/\mkern-7mu\big/}
{\Big}{\Big/\mkern-10mu\Big/}
{\bigg}{\bigg/\mkern-14mu\bigg/}
{\Bigg}{\Bigg/\mkern-18mu\Bigg/}
}
}
\ExplSyntaxOff
\begin{document}
$X\dslash H$
$X\dslash[\big] H$
$X\dslash[\Big] H$
$X\dslash[\bigg] H$
$X\dslash[\Bigg] H$
\end{document}

biggl/you are making/big. Corresponding spacing also will increase. Adjust\mkernvalue accordingly. But why should you usebiggl/, I don't understand. – Feb 23 '14 at 14:03\big, not\bigl. Also\biggl.and\biggr.are completely useless. – egreg Feb 23 '14 at 14:22XandHare replaced by something much larger. I do not see whybigglandbiggrare completely useless. Can you elaborate? – Jesko Hüttenhain Feb 23 '14 at 14:24\biggl.is doing? – egreg Feb 23 '14 at 14:58