Edit: Due to the comments of user119598 and 120579, the answer provided below may be incorrect (in some cases). If by "maximal commutative subring" the subring is not required to be a proper subset of $R$, then the original argument (below the dashed line) is indeed correct. However it seems standard to require that a maximal subring is required to be a proper subset, and for the rest of this edit I will be using this definition.
If $R$ is not commutative, then the original argument is again correct. In this case, the union of a chain of commutative subrings is commutative and hence can not be all of $R$.
If $R$ is commutative, then the argument is not correct as we can not guarantee the union is a proper subset as pointed out in the comments below. Actually, in this case the statement may be false. A maximal commutative subring is just a maximal subring of the commutative ring $R$. The abstract of the paper Most Commutative Rings Have Maximal Subrings by Azarang and Karamzadeh seems to indicate that there are commutative rings for which a maximal subring does not exist. Unfortunately my university does not have access to the journal and I have been unable to get my hands on the paper.
To show that every chain has a maximal element, given a chain you should be able to find (e.g. construct) the maximal element. Hint: If $R$ has a 1, this is similar to how one shows that every ideal is contained in a maximal ideal.
Once you have this candidate, you must show it satisfies the properties required: i.e. you must show it is commutative.