How do I perform a regression when I know there is a multiplicative relationship between the regression coefficients? Suppose I'm attempting to use linear regression but instead of
$$y_i = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \varepsilon_i,$$
for scientific reasons, I know that $β_2=\alpha\beta_1$ and $β_4=\alpha\beta_3$. So my formula is actually:
$$y_i = \beta_0 + \beta_1 x_1 + \alpha\beta_1 x_2 + \beta_3 x_3 + \alpha\beta_3 x_4 + \varepsilon_i$$
Note that the variable $\alpha$ is the same in both equations but unknown. How should I tackle something like this?

Since $\alpha$ is unknown, this reduces to a nonlinear regression with the model form:
$$y_i = \beta_0 + \beta_1 [x_1 + \alpha x_2] + \beta_2 [x_3 + \alpha x_4] + \varepsilon_i.$$
You can implement nonlinear regression using the nls function in R.  For a nonlinear model you will need to specify a starting point for your parameters in order to implement the iterative fitting-method.  A reasonable starting point for the parameters in this case could be obtained by fitting the unconstrained linear model (your first model) and then taking the starting values for the parameters to be:
$$\begin{align}
\beta_0^* &= \hat{\beta}_0, \\[12pt]
\beta_1^* &= \hat{\beta}_1, \\[12pt]
\beta_2^* &= \hat{\beta}_3, \\[6pt]
\alpha^* &= \frac{\hat{\beta}_2+\hat{\beta}_4}{\hat{\beta}_1+\hat{\beta}_3}. \\[6pt]
\end{align}$$
(The values on the right-hand-side in these equations would be estimates from your first model.)
