Get complete graph from set of vertices?

RelationGraph[UnsameQ, vertices, VertexLabels -> "Name"]    

Alternatively, you can use any of the following to get the same result:

Graph[UndirectedEdge @@@ Subsets[vertices, {2}], VertexLabels -> "Name"]
AdjacencyGraph[vertices, ConstantArray[1, {5,5}]-IdentityMatrix[5], VertexLabels -> "Name"]
SetProperty[VertexReplace[#, Thread[VertexList@# -> vertices]] &@ CompleteGraph[5],
  VertexLabels -> "Name"]

To change just the labels you can use:

CompleteGraph[5, VertexLabels -> {k_ :> vertices[[k]]}]

same picture

Using AdjacencyGraph:

AdjacencyGraph[vertices, 
 AdjacencyMatrix[CompleteGraph[Length[vertices]]]]

Another way is with AdjacencyGraph.

SimpleGraph[
 AdjacencyGraph[vertices, ConstantArray[1, Length[vertices] {1, 1}]],
 VertexLabels -> Automatic
]

With IGraph/M, you can zero out the matrix diagonal directly:

AdjacencyGraph[vertices, 
 IGZeroDiagonal@ConstantArray[1, Length[vertices] {1, 1}], 
 VertexLabels -> Automatic]