You can use tex4ebook to create the ePub or mobi ebooks. It uses tex4ht for the conversion. tex4ht doesn't support skak package out of the box, but it is easily configurable, so we can add the support ourselves.
There are two things which needs to be addressed: the chess figures in the text, and the chessboards. The chess figures can be converted as Unicode characters, the chessboards must be converted as images.
The figures use a special font, which needs to be configured in tex4ht. Save the following text as SkakNew.htf and put it to the directory with your document:
SkakNew 32 121
'' '' 32
'!' '' 33
'”' '' 34
'#' '' 35
'$' '' 36
'%' '' 37
'&' '' 38
'’' '' 39
'(' '' 40
')' '' 41
'*' '' 42
'+' '' 43
',' '' 44
'-' '' 45
'.' '' 46
'/' '' 47
'0' '' 48
'1' '' 49
'2' '' 50
'3' '' 51
'4' '' 52
'5' '' 53
'6' '' 54
'7' '' 55
'8' '' 56
'9' '' 57
':' '' 58
';' '' 59
'¡' '' 60
'=' '' 61
'¿' '' 62
'?' '' 63
'@' '' 64
'→' '' 65
'♗' '' 66
'↑' '' 67
'⊙' '' 68
'△' '' 69
'□' '' 70
'⇗' '' 71
'⇔' '' 72
'⊞' '' 73
'╳' '' 74
'♔' '' 75
'⟂' '' 76
'≪' '' 77
'♘' '' 78
'≫' '' 79
'Ⅱ' '' 80
'♕' '' 81
'♖' '' 82
'>' '' 83
'⊕' '' 84
'○' '' 85
'⇆' '' 86
'W' '' 87
'×' '' 88
'Y' '' 89
'Z' '' 90
'[' '' 91
'“' '' 92
']' '' 93
'ˆ' '' 94
'˙' '' 95
'‘' '' 96
'⧉' '' 97
'⌓' '' 98
'±' '' 99
'⦂' '' 100
'∓' '' 101
'⩲' '' 102
'⩱' '' 103
'h' '' I cannot find it, it is plus followed by minus
'i' '' minus followed by plus
'=' '' 106
'k' '' 107
'l' '' 108
'#' '' 109
'n' '' 110
'o' '' 111
'p' '' 112
'q' '' 113
'r' '' 114
's' '' 115
't' '' 116
'u' '' 117
'v' '' 118
'w' '' 119
'x' '' 120
'y' '' 121
SkakNew 32 121
The images for chess boards can be requested with the following configuration. Save it as skak.4ht and save it to your document's dir as well:
\NewConfigure{SkakBoard}{2}
\pend:def\showboard{\a:SkakBoard}
\append:def\showboard{\b:SkakBoard}
\Configure{SkakBoard}{\Picture+{}}{\EndPicture}
\endinput
It just patches command \showboard to include special tex4ht commands which convert enclosed content as images.
You can compile your document to mobi using the following comamand. Note that you need to have kindlegen installed:
tex4ebook -f mobi minimum.tex
The result is following, shown in Calibre:

To get better picture quality, you can try to add the following line to the skak.4ht file (before \endinput):
\Configure{Picture}{.svg}
It will request the graphics in the svg format:

One last note: I got invalid ePub file because you used \author after maketitle, you should put it before it.
Edit:
If you want to use more commands which print to one canvas, you need to enclose them in custom environment, whose contents will be converted as an image:
\documentclass[a4paper,11pt,oneside]{book}
\usepackage[utf8]{inputenc}
\usepackage{hyperref}
\hypersetup{
colorlinks=true,
linkcolor=[RGB]{41,41,41},
filecolor=magenta,
urlcolor=cyan,
pdftitle={Chess},
bookmarks=true,
}
\usepackage[ps]{skak}
\showmoverOn
\newenvironment{mymoves}{}{}
\begin{document}
\frontmatter
\title{My title}
\author{Sudheer}
\maketitle
\tableofcontents
\mainmatter
\chapter{Introduction}
\newgame
\mainline{1.e4 c5 2.Nf3 g6 3.Bc4 Bg7 4.Ng5 e6}
\begin{mymoves}
\showboard
\printarrow{a1}{a8}
\end{mymoves}
\end{document}
The important part here is:
\begin{mymoves}
\showboard
\printarrow{a1}{a8}
\end{mymoves}
You can configure the mymoves environment in the config file:
\Preamble{xhtml}
\Configure{Picture}{.svg}
\ConfigureEnv{mymoves}{\Picture*{}}{\EndPicture}{}{}
\begin{document}
\EndPreamble
Note that it seems that this sample needs to use the SVG output, it seems that dvipng doesn't support some PostScript.
The result:

skakpackage, but I know thatxskakhas been used for ePub generation usingtex4ebook: https://tex.stackexchange.com/q/307252/2891. – michal.h21 May 01 '17 at 15:20