The governing equation of transient heat transfer analysis is described as follows:
$$C \frac{dT}{dt}+K T = Q$$
When using backward difference scheme for the discretization of the time we get the following equation:
$$(\frac{C}{\Delta t}+K)T_n = Q_n+\frac{C}{\Delta t}T_{n-1}$$
Where $C$ is the heat capacity matrix, $K$ is the conductivity matrix, $T_n$ is the temperature for time step $n$, $T_{n-1}$ is the temperature for time step $n-1$ and $Q$ is a heat flux vector.
The term becomes nonlinear as soon as I implement the radiation boundary condition.
My question now is, which methods can be used to solve the above nonlinear problem? Any further reading would be helpful.
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The Newton-Raphson method can be used to solve non-linear systems of equations.
The first step is to write your system as a root finding problem:
$$ f(T_n) = \left( \frac{C}{\Delta t} + K \right) T_n - Q_n - \frac{C}{\Delta t} T_{n-1} = 0 $$
Taylor expand this equation about an initial guess $T^0_n$, keeping only the linear term: $$ f(T_n) \approx f(T^0_n) + \left.\frac{\partial f(T_n)}{\partial T_n}\right|_{T^0_n} \Delta T^0_n $$ where $\left.\frac{\partial f(T_n)}{\partial T_n}\right|_{T^0_n}$ is the Jacobian of $f(T_n)$ evaluated at $T^0_n$. Replace $f(T_n)$ with 0, since this is the end goal of the root finding equation we're trying to solve. Re-arranging,
$$ \left.\frac{\partial f(T_n)}{\partial T_n}\right|_{T^0_n} \Delta T^0_n = -f(T^0_n) $$ This gives us a linear equation we can use to solve for $\Delta T^0_n$, which tells us how much we need to adjust $T^0_n$ by in order to get a (hopefully) better estimate of $T_n$: $$ T^1_n = T^0_n + \Delta T^0_n $$ If $T^1_n$ is not sufficiently close to the solution (i.e. $\|f(T^1_n)\| > \tau$, for some tolerance $\tau$), then you can just repeat the procedure with the new guess $T^1_n$.
Note that there are some potential caveats with the convergence of the base Newton-Raphson method, so you may need to augment it with something like a line search in order to achieve convergence.
Also, there are many libraries which implement some variation of this procedure, as well as other methods for solving non-linear systems, so I would look to use one of these first before trying to implement your own.
