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Let $\left|\psi_1\right\rangle=\alpha_1\left|0\right\rangle+\beta_1\left|1\right\rangle$ , and $\psi_2=\alpha_2\left|0\right\rangle+\beta_2\left|1\right\rangle$.
Assume that $\left|\psi\right\rangle$ is a mixed state of $w_1*\left|\psi_1\right\rangle$ and $w_2*\left|\psi_2\right\rangle$ with $w_1+w_2=1$.
How to calculate the state vector of $\left|\psi\right\rangle$ ?
If the state is a mixed state of $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ (or any two states) it cannot be written as a state vector; state vectors only describe pure states, as opposed to mixed states.
If it is a coherent superposition of $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$, you can write:
$$\sqrt{w_{1}}|\psi_{1}\rangle + \sqrt{w_{2}}|\psi_{2}\rangle = (\sqrt{w_{1}}\alpha_{1} + \sqrt{w_{2}}\alpha_{2})|0\rangle + (\sqrt{w_{1}}\beta_{1} + \sqrt{w_{2}}\beta_{2})|1\rangle$$
The implicit assumption here is that $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ are orthogonal. If this is not the case, it makes less sense to sum the two states in such a way, but one can always normalize using their inner product $1+\sqrt{w_{1}w_{2}}|\langle\psi_{1}|\psi_{2}\rangle|^{2}$ (as you already pointed out in your comment):
$$ \frac{1}{1+\sqrt{w_{1}w_{2}}|\langle\psi_{1}|\psi_{2}\rangle|^{2}}\Big( \sqrt{w_{1}}|\psi_{1}\rangle + \sqrt{w_{2}}|\psi_{2}\rangle\Big) $$
If it is not a coherent superposition but indeed a (statistical) mixture of the two, a good (but wieldy) description of the state is:
"I am not certain what the state of the qubit is. It is either $|\psi_{1}\rangle$ or $|\psi_{2}\rangle$, but not a proper superposition of the two. With probability $w_{1}$ it is $|\psi_{1}\rangle$, and with probability $w_{2}$ it is $|\psi_{2}\rangle$."
Now, that is quite a mouthful and therefore we also have a different technique of writing down the state, making use of density matrices. The density matrix $\rho$ describing the state is
(with $\langle\psi_{1}|$ the complex transpose of $|\psi_{1}\rangle$):
$$ \begin{split} \rho &= w_{1}|\psi_{1}\rangle \langle\psi_{1}| + w_{2}|\psi_{2}\rangle \langle\psi_{2}| \\ &= w_{1} \begin{bmatrix}\alpha_{1}\alpha^{*}_{1} & \alpha_{1}\beta^{*}_{1} \\ \alpha^{*}_{1}\beta_{1} & \beta_{1}\beta_{1}^{*}\end{bmatrix} + w_{2} \begin{bmatrix}\alpha_{2}\alpha^{*}_{2} & \alpha_{2}\beta^{*}_{2} \\ \alpha^{*}_{2}\beta_{2} & \beta_{2}\beta_{2}^{*}\end{bmatrix} \\ \end{split} $$
For more info you could check, for instance, the Wikipedia page on density matrices or a good Quantum mechanics or quantum computing book.
