How to draw line connecting points on circle

A pretty easy solution is to use Graph because there you can

• create edges
• position the vertices on a circle
• label all vertices
• put a circle on it

A somewhat condensed form of this idea is the following

drawMe[l_List] :=
 With[{ed = Range[Length[l]], dphi = 2 Pi/(Length[l])},
  Graph[ed, (UndirectedEdge @@@ 
        Subsets[Flatten[#], {2}]) & /@ (Position[l, #] & /@ 
       Union[l]) // Flatten, 
   VertexLabels -> (Rule @@@ Transpose[{ed, l}]),
   VertexCoordinates -> 
    Table[{Cos[phi], Sin[phi]}, {phi, 0, 2 Pi - dphi, dphi}], 
   Epilog -> {Circle[]}, ImagePadding -> 15]]

drawMe[{1, 2, 3, 4, 5, 1, 2, 3, 5, 4}]

Btw, you can create nice things with this

drawMe@Table[Mod[i, 8], {i, 2^6}]

This is how I would do it:

coords[list_] := 
  With[{offset = 2 Pi/Length[list]}, Array[{Cos[# offset], Sin[# offset]} &, Length[list]]]
lines[list_] := Line[coords[list][[#]] & /@ Partition[Ordering[list], 2]]
labels[labels_, scaling_] := MapThread[Text, {labels, scaling coords[labels]}]

b = {1, 2, 3, 4, 5, 1, 2, 3, 5, 4};
Graphics[{
  Circle[],
  lines[b],
  labels[b, 1.1]
  }]

It assumes there are two of each label. It would have to be adjusted to work for other cases. I think the labeling function will work for your case as well. What it does is take the coordinates of the vertices and multiply them by some constant, to put them further away from the middle.

list = {1, 2, 3, 4, 5, 1, 2, 3, 5, 4};

Using RelationGraph

RelationGraph[UnsameQ[##] && SameQ @@ list[[{##}]] &, Range@Length@list, 
  VertexCoordinates -> CirclePoints[Length@list], 
  VertexLabels -> (MapIndexed[#2[[1]] -> Placed[#, Center] &, list]), 
  GraphStyle -> "SmallNetwork", Prolog -> {Red, Circle[]}]

Using AdjacencyGraph

am = Outer[Boole[Equal@##] &, list, list] - IdentityMatrix[Length@list];
AdjacencyGraph[am, VertexCoordinates -> CirclePoints[Length @ list], 
 VertexLabels -> (MapIndexed[#2[[1]] -> Placed[#, Center] &, list]), 
 GraphStyle -> "SmallNetwork", Prolog -> {Red, Circle[]}]

Note: If you replace CirclePoints[Length @ list]with

Table[{Cos[2 Pi k /Length[list]], Sin[2 Pi k /Length[list]]}, {k, Length@list}]

the second approach also works in versions before 10.1,