You have so many parameters that you cannot do it in a single plot. You can do it with a 3D graph

With[{K2 = \[Beta]*K1},
 Manipulate[
  Plot3D[
   (((\[Beta] - \[Alpha]) ((\[Beta] - 1) \[Alpha] - 
          2))/(2 (1 + \[Alpha])^2 (1 + \[Beta])))/K1,
   {\[Alpha], 0, 2}, {\[Beta], 2, 5}, 
   AxesLabel -> {\[Alpha], \[Beta], A}],
  {{K1, 1}, 0, 2}]]

or with a 2D graph

With[{K2 = \[Beta]*K1},
 Manipulate[
  Plot[
   (((\[Beta] - \[Alpha]) ((\[Beta] - 1) \[Alpha] - 
          2))/(2 (1 + \[Alpha])^2 (1 + \[Beta])))/K1,
   {\[Beta], 2, 5}, AxesLabel -> {\[Beta], A}],
  {\[Alpha], 0, 2}, {{K1, 1}, 0, 2}]]