You need to provide file SkakNew-FigurineBold.htf. This is virtual hypertext font, is used by tex4ht to convert characters from font encoding to xml encoding.
This file should look like this:
SkakNew-FigurineBold 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
'A' '' 65
'♗' '1' 66
'C' '' 67
'D' '' 68
'E' '' 69
'F' '' 70
'G' '' 71
'H' '' 72
'I' '' 73
'J' '' 74
'♔' '1' 75
'L' '' 76
'M' '' 77
'♘' '1' 78
'O' '' 79
'P' '' 80
'♕' '1' 81
'♖' '1' 82
'S' '' 83
'T' '' 84
'U' '' 85
'V' '' 86
'W' '' 87
'X' '' 88
'Y' '' 89
'Z' '' 90
'[' '' 91
'“' '' 92
']' '' 93
'ˆ' '' 94
'˙' '' 95
'‘' '' 96
'a' '' 97
'b' '' 98
'c' '' 99
'd' '' 100
'e' '' 101
'f' '' 102
'g' '' 103
'h' '' 104
'i' '' 105
'j' '' 106
'k' '' 107
'l' '' 108
'm' '' 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-FigurineBold 32 121
Important lines for you are
'♗' '1' 66
'♔' '1' 75
'♘' '1' 78
'♕' '1' 81
'♖' '1' 82
these lines means that characters B, N, P, Q and R are converted as xml entities for unicode representation of chess characters. Number 1 in second column means, that these characters will be converted to images. This is needed in your case, because in ebooks you don't have much control about fonts. Particularly older versions of kindle doesn't support chess unicode characters, so you should insert them as images. You also should create custom config file for tex4ht conversion scripts, because by default, images are suited for computer screen with lower resolution, but e-ink devices have resolution about 167 ppi. For more info about this, see this answer. If you want characters instead of images, just delete these 1 on these lines.
You should also create config file for skak, because if you try to print chessboard, instead of image, you will also get only some characters.
File skak.4ht
\NewConfigure{showboard}{2}
\let\oldshowboard\showboard
\Configure{showboard}{\Picture*{}}{\EndPicture}
\renewcommand\showboard{\a:showboard\oldshowboard\b:showboard}
Some sample file skak-sample.tex:
\documentclass{book}
\usepackage[ps,mover]{skak}
\begin{document}
\parindent=0pt
\newgame
\mainline{1. d4 d5} \mainline{2. c4 Nf6 }
\mainline{3.Ka1 Qa1 4.Ra1 Ba1 5.Na1 a1}
\showboard
\end{document}
Now with htlatex skak-sample you will get:
