Diagonal lines in table cell

Here's a solution using TikZ:

documentclass[11pt]{article}
usepackage[T1]{fontenc}
usepackage{tikz}
usetikzlibrary{matrix}

begin{document}

begin{tikzpicture}%[thick]
  matrix (mat) [%
    matrix of nodes,
    nodes in empty cells,
    text width=0.8cm,
    text height=10pt,
    text depth=2pt,
    text badly centered
  ] 
  {%
     & & & & 20 
     & & & & 30  
     & & & & 45  
    15 & 12 & 18 & 50 & raisebox{5pt}{$a_i$}hspace{-15pt}llap{raisebox{-1pt}{$b_j$}}
  };
  % horizontal lines
  foreach i in {1,2,3,4}
        draw (mat-i-1.north west) -- (mat-i-5.north east);
  draw (mat-4-1.south west) -- (mat-4-5.south east);
  % vertical lines
  foreach j in {1,2,3,4,5}
    draw (mat-1-j.north west) -- (mat-4-j.south west);
  draw (mat-1-5.north east) -- (mat-4-5.south east);
  % diagonal line    
  draw (mat-4-5.north west) -- (mat-4-5.south east);
end{tikzpicture}

end{document}

EDIT: in the solution I initially gave, the last cell contents was not what the OP wanted. I corrected it.

It might be possible to draw a diagonal line which fits exactly in a table cell, but it might be easier to draw the whole table as a picture.

Here my attempt using tikz. For large tables the need for nodes for each cell might be quite an effort, but it should be OK for smaller ones.

documentclass[11pt]{article}
usepackage[T1]{fontenc}
usepackage{tikz}
begin{document}
begin{tikzpicture}[x=.75cm,y=.5cm]
    draw (0,0) grid [step=1] (5,4);
    node at (0.5,0.5) {15};
    node at (1.5,0.5) {12};
    node at (2.5,0.5) {18};
    node at (3.5,0.5) {50};
    node at (4.5,3.5) {20};
    node at (4.5,2.5) {30};
    node at (4.5,1.5) {45};
    draw (4,1) -- (5,0);
    node at (5.0,1.0) [below left,inner sep=1pt] {small$a_i$};
    node at (4.0,0.0) [above right,inner sep=1pt] {small$b_j$};
end{tikzpicture}
end{document}

Exact solution with TikZ:

documentclass[11pt]{article}
usepackage[T1]{fontenc}
usepackage{array}
usepackage{makecell}
newcolumntype{x}[1]{>{centeringarraybackslash}p{#1}}

usepackage{tikz}
newcommanddiag[4]{%
  multicolumn{1}{p{#2}|}{hskip-tabcolsep
  $vcenter{begin{tikzpicture}[baseline=0,anchor=south west,inner sep=#1]
  path[use as bounding box] (0,0) rectangle (#2+2tabcolsep,baselineskip);
  node[minimum width={#2+2tabcolsep},minimum height=baselineskip+extrarowheight] (box) {};
  draw (box.north west) -- (box.south east);
  node[anchor=south west] at (box.south west) {#3};
  node[anchor=north east] at (box.north east) {#4};
 end{tikzpicture}}$hskip-tabcolsep}}

begin{document}
setlength{extrarowheight}{0.1cm}
begin{tabular}{|x{0.5cm}|x{0.5cm}|x{0.5cm}|x{0.5cm}|x{0.5cm}|}hline
&&&&20 hline
&&&&30 hline
&&&&45 hline
15&12&18&50&diag{.1em}{.5cm}{$a_i$}{$b_j$} hline
end{tabular}
end{document}

Also it is a reimplementation of diaghead.

Here's another solution using Tikz and based on a given solution by Heiko Oberdiek.

documentclass{article}
pagestyle{empty}% for cropping
usepackage{array}
usepackage{makecell}
newcolumntype{x}[1]{>{centeringarraybackslash}p{#1}}

usepackage{tikz}
usetikzlibrary{calc}
usepackage{zref-savepos}

newcounter{DiagonalizedEntry}
renewcommand*{theDiagonalizedEntry}{NTE-thevalue{DiagonalizedEntry}}

newcommand*{diagonalize}[2]{%
  multicolumn{1}{@{}c@{}|}{%
    stepcounter{DiagonalizedEntry}%
    vadjust pre{zsavepos{theDiagonalizedEntry t}}% top
    vadjust{zsavepos{theDiagonalizedEntry b}}% bottom
    zsavepos{theDiagonalizedEntry l}% left
    hspace{0pt plus 1filll}%
    zsavepos{theDiagonalizedEntry r}% right
    tikz[overlay]{%
      draw[red]
        let
          n{llx}={zposx{theDiagonalizedEntry l}sp-zposx{theDiagonalizedEntry r}sp}, % x left
          n{urx}={0}, % x right
          n{lly}={zposy{theDiagonalizedEntry b}sp-zposy{theDiagonalizedEntry r}sp}, % y bottom
          n{ury}={zposy{theDiagonalizedEntry t}sp-zposy{theDiagonalizedEntry r}sp} %  y top
    in
        (n{llx}, n{ury}) -- (n{urx}, n{lly})
    node[anchor=south west] at (n{llx}, n{lly}) {#1}
    node[anchor=north east] at (n{urx}, n{ury}) {#2}
    ;
    }% 
  }%
}

begin{document}
renewcommand{arraystretch}{2}
renewcommandtabcolsep{10pt}
begin{tabular}{|x{0.5cm}|x{0.5cm}|x{0.5cm}|x{0.5cm}|x{0.5cm}|}hline
&&&&20 hline
&&&&30 hline
&&&&45 hline
15&12&18&50&diagonalize{$a_i$}{$b_j$}  hline
end{tabular}
end{document}

Okay as this seems to be the question that all duplicates link to eventually, I want to include @Leo Liu's solution from here as well as this seems to solve the problem in the easiest way.

In short: Use the @Leo Liu's package diagbox.

MWE (coming from a different context and does not exactly answer to this question) shamelessly copied from Leo's answer (link above):

documentclass{article}

usepackage{diagbox}

begin{document}

begin{tabular}{|l|c|c|}hline
diagbox[width=10em]{DiagColumn Head I}{Diag ColumnHead II}&
  Second & Third  hline
& foo & bar  hline
end{tabular}

end{document}

Resulting in

If you like this solution make sure you go ahead and show Leo's original answer some love.

---

Admittedly this solution doesn't work so well (out of the box) with the provided MWE but if one uses normal column types instead of the custom ones, one gets a really good result:

documentclass{standalone}
usepackage{diagbox}

begin{document}
    begin{tabular}{|c|c|c|c|c|}
        hline
           &    &    &    &           20            hline
           &    &    &    &           30            hline
           &    &    &    &           45            hline
        15 & 12 & 18 & 50 & diagbox{$a_i$}{$b_j$}  hline
    end{tabular}
end{document}

Results in

A solution with {NiceTabular} of nicematrix which has a built-in command diagbox. Moreover, you can draw all the rules with only one key hvlines.

documentclass{article}
usepackage{nicematrix}

begin{document}
setlength{extrarowheight}{1mm}
begin{NiceTabular}{ccccc}[hvlines]
   &    &    &    &           20            
   &    &    &    &           30            
   &    &    &    &           45            
15 & 12 & 18 & 50 & diagbox{$a_i$}{$b_j$}  
end{NiceTabular}
end{document}

Remark: For versions of nicematrix prior to 5.0, you have to put {CCCCC} instead of {ccccc}.