Adding complex phonetic symbols to LaTeX

As explained by this answer, you can either use tipa and compile using pdfLaTeX or load a Unicode IPA font using fontspec and compile using XeLaTeX or LuaLaTeX. I strongly recommend using a Unicode IPA font rather than tipa for the reasons outlined in this answer.

Here is an example that uses the IPA versions of most of the symbols shown in the image you posted (I included both the palato-alveolar affricate and palatal stop symbols). I entered these symbols directly in my TeX editor using an IPA keyboard layout, but you could also use one of many online IPA pickers and copy and paste from there into your editor. I will leave the table formatting up to you.

\documentclass{article}

\usepackage{fontspec}
\setmainfont{Charis SIL}

\begin{document}
\noindent p t̪ t ʈ t͡ʃ c k\\
pʰ t̪ʰ ʈʰ t͡ʃʰ cʰ kʰ \\
b d̪ ɖ d͡ʒ ɟ ɡ \\
bʱ d̪ʱ ɖʱ d͡ʒʱ ɟʱ ɡʱ \\
f s ʂ ʃ h \\
m n̪ n ɳ ɲ ŋ \\
r ɽ ͏ɻ\\
l ɭ \\
w v j
\end{document}

As shown below, the font you select does make a difference in whether your symbols will come out right. Some fonts simply don't have the glyphs for most IPA symbols (although the ones shown here do have all the glyphs in the example), and others do a poor job of stacking diacritics and placing things like the tie bar used in affricates. So choose your font wisely, also paying attention to how well it does with other formatting you need such as bold and small caps.

Charis SIL:

Brill:

Linux Libertine O:

Times New Roman:

As was told you in the comments, you can (and should if you're using xelatex) use a font that supports all IPA glyphs.

Since you're using a table, you can automatize this so that only the cells with IPA use that font, and the headers use the regular font. For the headers I'm using a particular font here just to show the difference, but you can use anything you like.

Output

Code

\documentclass{article}
\usepackage[margin=2.5cm]{geometry}
\usepackage{fontspec}
\usepackage{booktabs}
\usepackage{array}
\setmainfont{Century Gothic}
\newfontfamily\ipafont{Charis SIL}
\newcommand\ipa[1]{{\ipafont #1}}

% To keep the header with normal font
\makeatletter
\newcommand*{\rowstyle}[1]{% sets the style of the next row
  \gdef\@rowstyle{\leavevmode#1}%
  \@rowstyle\ignorespaces}
\newcolumntype{=}{>{\gdef\@rowstyle{}}}
\newcolumntype{+}{>{\@rowstyle}}
\makeatother

% Column type with ipa font
\newcolumntype{A}{+>{\ipafont}l}

\begin{document}
\begin{tabular}{=l*{7}{A}}
    \toprule
    \rowstyle{\normalfont}
    ~                   & Labial & Dental & Alveolar & Retroflex & Palatal & Velar & Glottal\\ \midrule
    Stop                & ~      & ~      & ~        & ~         & ~       & ~     & ~       \\
    Voiceless           & ~      & ~      & ~        & ~         & ~       & ~     & ~       \\
    Voiceless aspirated & ~      & ~      & ~        & ~         & ~       & ~     & ~       \\
    Voiced              & ~      & ~      & ~        & ~         & ~       & ~     & ~       \\
    Voiced aspirated    &      & ~      & ~        & ~         & ~       & ~     & ~       \\
    Fricative           &    & ~      & ~        & ~         & ~       & ~     & ~       \\
    Nasal               & t͡ʃʰ    & ~      & ~        & ~         & ~       & ~     & ~       \\
    \bottomrule
    \end{tabular}
\end{document}