If you change the interval to $[-1,1]$ instead of $[0,1]$, the equivalent inner product would be $$\tag{$*$} (f,g) = (f,g)_{L^2([-1,1])} + \lambda (f',g')_{L^2([-1,1])},$$ where $\lambda = 2$.

One way to get an orthogonal basis is to start with polynomials and apply Gram-Schmidt orthogonalization to them. You get what might be called Sobolev-Legendre polynomials. If you search for _Sobolev orthogonal polynomials_ you find lots of references, of which [MX] seems to be a reasonable review. Digging a bit deeper, one finds [M, Thm.3.3] an explicit recurrence relation for Sobolev-Legendre polynomials $S^\lambda_n(x)$, normalized by $S^\lambda_n(1) = 1$, with respect to the inner product $(*)$:

$$
  S^\lambda_n(x) = S^\lambda_{n-2} + a_n (P_n(x) - P_{n-2}(x)) ,
$$
where $P_n$ are the usual [Legendre polynomials](https://en.wikipedia.org/wiki/Legendre_polynomials) and
$$
  a_n = \sum_{k=0}^{[\frac{n-1}{2}]} \left(\frac{\lambda}{4}\right)^k
    \frac{1}{(2n)!} \frac{(n+2k-1)!}{(n-2k-1)!}
$$

[MX] <cite authors="Francisco Marcellán and Yuan Xu" mrnumber="3360352" cite="_Expo. Math._ **33** (2015), no. 3, 308--352">_Francisco Marcellán and Yuan Xu_, MR 3360352 [**On Sobolev orthogonal polynomials**](http://dx.doi.org/10.1016/j.exmath.2014.10.002), _Expo. Math._ **33** (2015), no. 3, 308--352.</cite>

[M] <cite authors="H. G. Meijer" cite="_Niew Arch. Wisk._ **14** (1996), 93--112">_H. G. Meijer_, [**A short history of orthogonal
polynomials in a Sobolev space I. The non-discrete case**](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.28.5540&rep=rep1&type=pdf), _Niew Arch. Wisk._ **14** (1996), 93--112.</cite>