High entropy alloy in TikZ

I think the simplest way is to define your colors in a list with \pgfmathdeclarerandomlist and then you can randomly select an item using \pgfmathrandomitem. Since you have a 2D TikZ image, changing the view means to change the position (coordinates) of the balls.

Code:

\documentclass[tikz]{standalone}

\pgfmathdeclarerandomlist{colors}{%
    {red}%
    {green}%
    {blue}%
    {yellow}%
}

\begin{document}
\begin{tikzpicture}[]
  \foreach \i in {1,...,12} {
      \foreach \j in {1,...,6} {
          \foreach \k in {1,...,4} {
              \pgfmathrandomitem{\randColor}{colors} 
              \shade[ball color=\randColor] (\i-\j/3, {0.5*\j+\k}) circle(0.4);
            }
        }
    }
\end{tikzpicture}
\end{document}

By the way, the answers to Drawing 3d crystal lattice with molecular layer in tikz show you other possibilities to build such a cube of "balls". These can be also randomly colored. For instance, for the answer of JLDiaz:

\documentclass{standalone}
\usepackage{tikz}

\usetikzlibrary{calc,fadings,decorations.pathreplacing}

\pgfmathdeclarerandomlist{colors}{%
    {red}%
    {green}%
    {blue}%
    {yellow}%
}

\begin{document}

% You can tweak these
\def\ballradius{0.45}
%

\def\DrawRow#1#2{
    \foreach \x in {0,...,#2}
    \pgfmathrandomitem{\randColor}{colors}
       \shade[ball color=\randColor] ($(#1) +(\x, 0,0)$) circle(\ballradius);
}
\def\DrawOddPlane#1{ 
  \pgfmathsetmacro{\aux}{#1-1}
  \foreach \z in {0,...,#1} {
      \DrawRow{0,0,\z}{#1}
      \if\z#1\relax\else
      \DrawRow{0.5,0,\z+0.5}{\aux}
      \fi
  }
}
\def\DrawEvenPlane#1{ 
  \pgfmathsetmacro{\aux}{#1-1}
  \foreach \z in {0,...,#1} {
      \DrawRow{0.5,0,\z}{\aux}
      \if\z#1\relax\else
      \DrawRow{0,0,\z+0.5}{#1}
      \fi
  }
}

\begin{tikzpicture}
   \foreach \y in {0,...,3} {
      \begin{scope}[yshift=\y cm]
          \DrawOddPlane{3}
      \end{scope}
      \if\y3\relax\else
      \begin{scope}[yshift=\y cm + 0.5cm]
          \DrawEvenPlane{3}
      \end{scope}
      \fi
  }
    \pgfmathsetmacro{\cubex}{1}
    \pgfmathsetmacro{\cubey}{1}
    \pgfmathsetmacro{\cubez}{1}
    \draw (3,3,3) -- ++(-\cubex,0,0) -- ++(0,-\cubey,0) -- ++(\cubex,0,0) -- cycle;
    \draw (3,3,3) -- ++(0,0,-\cubez) -- ++(0,-\cubey,0) -- ++(0,0,\cubez) -- cycle;
    \draw (3,3,3) -- ++(-\cubex,0,0) -- ++(0,0,-\cubez) -- ++(\cubex,0,0) -- cycle;
\end{tikzpicture}
\end{document} 

With the great help of @Ñako, here's the final result.

% Cartoon of the AlCoCrFeNi high entropy alloy (HEA) with body-centered cubic (BCC) lattice.

\documentclass[tikz]{standalone}

\pgfmathdeclarerandomlist{colors}{{red!80}{teal}{blue!80}{orange}{blue!20}}

\begin{document}
\begin{tikzpicture}
  \foreach \i in {1,...,12} {
      \foreach \j in {1,...,4} {
          \foreach \k in {1,...,4} {
              \pgfmathrandomitem{\randColor}{colors}
              \shade[ball color=\randColor] (-\i+0.3*\j, -0.2*\j+1.2*\k) circle(0.3);
            }
          \foreach \k in {1,...,3} {
              \pgfmathrandomitem{\randColor}{colors}
              \shade[ball color=\randColor] (-\i+0.5+0.3*\j, -0.2*\j+1.2*\k+0.6) circle(0.3);
            }
        }
    }
  \foreach \el/\color [count=\n] in {Al/red!80, Co/blue!80, Cr/teal, Fe/orange, Ni/blue!20} {
      \shade[ball color=\color] (2, 5.5-\n) circle(0.3) node[right=1em] {\el};
    }
\end{tikzpicture}
\end{document}