How to draw a line that passes through rows and columns of a table?

With only basic tools:

\documentclass{article}
\usepackage{float}
\usepackage[table, svgnames]{xcolor}
\usepackage{hhline, array}

\begin{document}

\begin{table}[H]
\centering\setlength{\extrarowheight}{1pt}
\begin{tabular}{|c|c|c|}
\hhline{|---|}
\noalign{\vskip \dimexpr1.5ex + 1pt-0.4pt\relax}
\hhline{>{\arrayrulecolor{Crimson}}--->{\arrayrulecolor{black}}|}
\noalign{\vskip\dimexpr-1.5ex- 1pt\relax}
    a & b & c \\
    \hhline{|---|}
    d & e & f \\
    \hhline{|---|}
    h & i & j \\
    \hhline{|---|}
\end{tabular}

\end{table}

\end{document} 

One possibility is to use TikZ and tikzmark.

\documentclass{article}
\usepackage{float}
\usepackage{tikz}
\usetikzlibrary{tikzmark}
\begin{document}

\begin{table}[H]
    \centering
    \begin{tabular}{|c|c|c|}
\hline
        \tikzmarknode{a}{a} & b & \tikzmarknode{c}{c} \\
        \hline
        d & e & f \\
        \hline
        h & i & j \\
        \hline
    \end{tabular}%
\begin{tikzpicture}[overlay,remember picture]
 \draw[thick,red,-stealth] ([xshift=-1ex]a.west) -- ([xshift=1ex]c.east);
\end{tikzpicture}%  

\end{table}
\end{document}

Note that the syntax of \tikzmarknode is \tikzmarknode{<id>}{<content>}, where id is an identifier and content is the content. So, if you want a 1, say instead of the a, you could use the same identifier and only change the second argument, \tikzmarknode{a}{1}. Then there will be a 1 in the table.

\documentclass{article}
\usepackage{float}
\usepackage{tikz}
\usetikzlibrary{tikzmark}
\begin{document}

\begin{table}[H]
    \centering
    \begin{tabular}{|c|c|c|}
\hline
        \tikzmarknode{a}{1} & 2 & \tikzmarknode{c}{3} \\
        \hline
        \tikzmarknode{goat}{d} & e & f \\
        \hline
        \tikzmarknode{duck}{h} & i & j \\
        \hline
    \end{tabular}%
\begin{tikzpicture}[overlay,remember picture]
 \path[left color=blue,right color=red] 
  ([xshift=-1ex,yshift=-0.3pt]a.west) rectangle ([xshift=1ex,yshift=0.3pt]c.east);
 \draw[red,shorten >=-0.2ex,shorten <=-0.2ex]  (goat.north) -- (duck.south);
\end{tikzpicture}%  

\end{table}
\end{document}

This example is also to illustrate that with TikZ you can go far beyond monochrome lines.