Let you have a polynomial

    f[x_] := 1 - x + 2 x^2 - 5 x^4
    n = 5; (* Polynomial order + 1 *)
    
Now you have to solve a system of linear equations
    
    a = Transpose@PadRight@Table[CoefficientList[ChebyshevT[k - 1, x], x], {k, n}];
    c = LinearSolve[a, CoefficientList[f[x], x]]
    (* {1/8, -1, -3/2, 0, -5/8} *)
    
    Sum[c[[k]] ChebyshevT[k - 1, x], {k, n}] // Expand
    (* 1 - x + 2 x^2 - 5 x^4 *)

If the numerical values of coefficients are enough you can apply extremly fast method with Fourier transform since

$$
T_n(x) = \cos(n \arccos x)
$$

    c = 2 MapAt[#/2 &, #, 1]/Sqrt[n] &@FourierDCT@f@Cos[π Range[.5, n]/n]
    (* {0.125, -1., -1.5, -9.93014*10^-17, -0.625} *)
    
    Sum[c[[k]] ChebyshevT[k - 1, x], {k, n}] // Expand // Chop
    (* 1. - 1. x + 2. x^2 - 5. x^4 *)

P.S. The last method can be used for Chebyshev expansion of arbitrary function (there is a problem of Gibbs oscillations, but it is another story).