Given a smooth manifold $M$ of dimension $d$, consider the frame bundle $FM \overset{\pi_{FM}}\longrightarrow M$. We can construct the tangent bundle $TM \overset{\pi_{TM}}\longrightarrow M$ as an associated $GL(d, \mathbf{R})$-bundle by first taking the product $FM \times \mathbf{R}^d$. Intuitively, this is a bundle over $M$ with typical fiber $GL(d, \mathbf{R}) \times \mathbf{R}^d$. Next, we take a quotient by $\sim_{GL(d, \mathbf{R})}$, where $\sim_{GL(d, \mathbf{R})}$ is given by $$\left([\mathbf{e}_i], v\right) \sim \left([\mathbf{e}_i] \blacktriangleleft g, g^{-1} \blacktriangleright v\right)$$ for all $g \in GL(d, \mathbf{R})$, where $[\mathbf{e}_i]$ denotes frames and $v$ denotes coefficients. Letting the quotient space be denoted $(FM \times \mathbf{R}^d) / GL(d, \mathbf{R})$, there exists a vector bundle isomorphism $(FM \times \mathbf{R}^d)/GL(d, \mathbf{R}) \longrightarrow TM$ given by $[\left[\mathbf{e}_i\right], v] \mapsto \sum_i v_i\mathbf{e}_i$.
Analogously, I would like to construct a general tensor bundle $T^p_qM$ as an associated $GL(d, \mathbf{R})$-bundle. I suspect that this is done as follows: Letting $\rho: GL(d, \mathbf{R}) \longrightarrow GL(d, \mathbf{R})$ be a group representation, construct the associated vector bundle $(FM \times \mathbf{R}^c)/\sim_{\rho}$, where $\sim_{\rho}$ is given by $$ \left([\mathbf{e}_i], t\right) \sim \left([\mathbf{e}_i] \blacktriangleleft g, \rho(g^{-1}) \blacktriangleright t\right) $$ for all $g\in GL(d, \mathbf{R})$, where $\mathbf{R}^c \cong T_pM^{\otimes p}\otimes T^*_pM^{ \otimes q}$ and $t$ denotes tensor coefficients. My understanding is that $\rho$ should encode the typical transformation law associated to a $(p, q)$-tensor under a change of frame. (Q1) Is this the correct $\rho$? Additionally, I have seen the transformation law for a $(p,q)$-tensor given by $$ \widetilde{T}^{i_1,\ldots,i_p}_{j_1,\ldots,j_q} = \sum_{\substack{1\leq k_1,\ldots, k_p \leq d}\\1\leq\ell_1,\ldots,\ell_q\leq d}T^{ k_1,\ldots, k_p }_{\ell_1,\ldots,\ell_q}g_{i_1k_1}\cdots g_{i_pk_p}g^{j_1\ell_1}\cdots g^{j_q\ell_q}. $$ However, I'm having trouble seeing how such a transformation can be described using a representation $\rho$. (Q2) I was under the assumption that the tensor components live in a multidimensional array, so how can this live in the matrix group $GL(d, \mathbf{R})$?