ColorFunction based on array index in ListLinePlot

data = Table[E^-0.001 x {Cos[x], Sin[x]}, {x, 0, 100, 0.1}];

indexlist = Rescale[Range @ Length @ data] (* or your external list to control color*);

ListLinePlot

iF = Interpolation[MapThread[{#, #2} &, {data, indexlist}], InterpolationOrder -> 1];

ListLinePlot[data,
 AspectRatio -> 1, 
 ColorFunctionScaling -> False, 
 ColorFunction -> (ColorData[{"Rainbow", "Reversed"}][ iF[#, #2]]&)]

Alternative methods:

PolarPlot

PolarPlot[E^-0.001 x , {x, 0, 100}, 
 ColorFunction -> (ColorData[{"Rainbow", "Reversed"}][#3] &)

same picture

ParametricPlot

ParametricPlot[E^-0.001 x {Cos[x], Sin[x]}, {x, 0, 100}, 
 ColorFunction -> (ColorData[{"Rainbow", "Reversed"}][#3] &)]

same picture

Graphics + VertexColors

Graphics[Line[data,  VertexColors -> (ColorData[{"Rainbow", "Reversed"}] /@ indexlist)], 
  Axes -> True]

same picture

I think it is easier to generate the plot you want with ParametricPlot Like so:

With[{n = 100},
  ParametricPlot[E^-0.001 x {Cos[x], Sin[x]}, {x, 0, n},
    Mesh -> n - 1,
    MeshStyle -> Transparent,
    MeshShading ->Table[ColorData[L{"Rainbow", "Reverse"}][i/n], {i, n}]]]

With[{lines=Line/@Partition[Table[E^-0.001 x {Cos[x], Sin[x]}, {x, 0, 100, 0.1}],2,1]}, 
Graphics[MapIndexed[{ColorData[{"Rainbow","Reverse"}][First@#2/Length[lines]],#1}&,lines],
Frame -> True]]