How to create circle checkbox multiple choice exam with choice letters inside circle

Using some Tikz trickery it can be done. If you are using oneparcheckboxes, all you have to do is modify the checkboxchar and checkedchar commands:

% Make checkbox character a circle with the letter
checkboxchar{tikz[baseline={([yshift=-.8ex]current bounding box.center)}]node[shape=circle,minimum size=8mm,draw] at (0,0) {thechoice};}
% Make checked box character bold WITH surd
%checkedchar{tikz[baseline={([yshift=-.8ex]current bounding box.center)}]node[shape=circle,minimum size=8mm,draw] at (0,0) {} node at (0,0) {thechoicellap{$surd$}};}
% Make checked box character bold
checkedchar{tikz[baseline={([yshift=-.8ex]current bounding box.center)}]node[shape=circle,minimum size=8mm,draw] at (0,0) {} node at (0,0) {thechoice};}

But this will not work for the checkboxes environment because it does not keep track of the items using a counter. For that, you have to modify the environment:

makeatletter
    renewenvironment{checkboxes}%
       {setcounter{choice}{0}list{checkbox@char}%
          {%
            settowidth{leftmargin}{W.hskiplabelsephskip 2.5em}%
            defchoice{%
              if@correctchoice
                color@endgroup endgroup
              fi
              stepcounter{choice}
              item[checked@char]
              do@choice@pageinfo
            } % choice
            defCorrectChoice{%
              if@correctchoice
                color@endgroup endgroup
              fi
              ifprintanswers
                % We can't say choice here, because that would
                % insert an endgroup.
                % 2016/05/10: We say color@begingroup in addition to
                % begingroup in case CorrectChoiceEmphasis involves color
                % and the text exactly fills the line (which would
                % otherwise create a blank line after this choice):
                % 2016/05/11: We leave hmode if we're in it,
                % i.e., if there's no blank line preceding this
                % CorrectChoice command.  (Without this, the
                % special created by a color{whatever} command that might
                % be inserted by CorrectChoice@Emphasis would be appended 
                % to the previous choice, which could cause an extra
                % (blank) line to be inserted before this CorrectChoice.)
                % Since par and endgraf seem to cancel @totalleftmargin
                % (for reasons I don't understand), we'll do the following:
                % Motivated by  the def of leavevmode, 
                %      defleavevmode{unhboxvoidb@x}
                % we will now leave hmode (if we're in hmode):
                ifhmode unskipunskipunvboxvoidb@x fi
                begingroup color@begingroup @correctchoicetrue
                CorrectChoice@Emphasis
                stepcounter{choice}
                item[checked@char]
              else
                stepcounter{choice}
                item[checked@char]
              fi
              do@choice@pageinfo
            } % CorrectChoice
            letcorrectchoiceCorrectChoice
            labelwidthleftmarginadvancelabelwidth-labelsep
            topsep=0pt
            partopsep=0pt
            checkboxeshook
          }%
       }%
       {if@correctchoice color@endgroup endgroup fi endlist}
     makeatother

And now for a MWE:

documentclass[a4paper,12pt,answers]{exam}

usepackage{tikz}

makeatletter
renewenvironment{checkboxes}%
   {setcounter{choice}{0}list{checkbox@char}%
      {%
        settowidth{leftmargin}{W.hskiplabelsephskip 2.5em}%
        defchoice{%
          if@correctchoice
            color@endgroup endgroup
          fi
          stepcounter{choice}
          item[checked@char]
          do@choice@pageinfo
        } % choice
        defCorrectChoice{%
          if@correctchoice
            color@endgroup endgroup
          fi
          ifprintanswers
            % We can't say choice here, because that would
            % insert an endgroup.
            % 2016/05/10: We say color@begingroup in addition to
            % begingroup in case CorrectChoiceEmphasis involves color
            % and the text exactly fills the line (which would
            % otherwise create a blank line after this choice):
            % 2016/05/11: We leave hmode if we're in it,
            % i.e., if there's no blank line preceding this
            % CorrectChoice command.  (Without this, the
            % special created by a color{whatever} command that might
            % be inserted by CorrectChoice@Emphasis would be appended 
            % to the previous choice, which could cause an extra
            % (blank) line to be inserted before this CorrectChoice.)
            % Since par and endgraf seem to cancel @totalleftmargin
            % (for reasons I don't understand), we'll do the following:
            % Motivated by  the def of leavevmode, 
            %      defleavevmode{unhboxvoidb@x}
            % we will now leave hmode (if we're in hmode):
            ifhmode unskipunskipunvboxvoidb@x fi
            begingroup color@begingroup @correctchoicetrue
            CorrectChoice@Emphasis
            stepcounter{choice}
            item[checked@char]
          else
            stepcounter{choice}
            item[checked@char]
          fi
          do@choice@pageinfo
        } % CorrectChoice
        letcorrectchoiceCorrectChoice
        labelwidthleftmarginadvancelabelwidth-labelsep
        topsep=0pt
        partopsep=0pt
        checkboxeshook
      }%
   }%
   {if@correctchoice color@endgroup endgroup fi endlist}
 makeatother

% Make checkbox character a circle with the letter
checkboxchar{tikz[baseline={([yshift=-.8ex]current bounding box.center)}]node[shape=circle,minimum size=8mm,draw] at (0,0) {thechoice};}
% Make checked box character bold WITH surd
%checkedchar{tikz[baseline={([yshift=-.8ex]current bounding box.center)}]node[shape=circle,minimum size=8mm,draw] at (0,0) {} node at (0,0) {thechoicellap{$surd$}};}
% Make checked box character bold
checkedchar{tikz[baseline={([yshift=-.8ex]current bounding box.center)}]node[shape=circle,minimum size=8mm,draw] at (0,0) {} node at (0,0) {thechoice};}


begin{document}
begin{questions}

question
Blah blah

begin{checkboxes}choice one
choice two
CorrectChoice three
end{checkboxes}

question
Blah blah

begin{oneparcheckboxes}
choice one
choice two
CorrectChoice three
end{oneparcheckboxes}

end{questions}
end{document}

This will shows as: