Firstly: The Ruling of the Sermon after the Eclipse Prayer
Scholars are of two positions regarding the ruling of delivering a sermon after the eclipse prayer:
The first position: No sermons are legislated after the eclipse prayer. This is the position of the majority: the Hanafis, Malikis,  However, it is mustahabb according to them for the imam admonish and remind people after the prayer. He encourages them to give charity, free slaves, and fast. They distinguish between admonition and a sermon. and Hanbalis.
The second position: A sermon is legislated after the eclipse prayer. This is the position of the Shafi`i school, a position within the Hanafi school, a narration from Ahmad, and the position of the majority of the Salaf and most imams of Hadith. It is the position of choice of Ishaq, al-Tabari, Ibn Baz, and Ibn `Uthaymin.
Secondly: How to Deliver a Sermon for the Eclipse Prayer
Scholars who hold the position that a sermon is legislated after the eclipse prayer hold two positions regarding how it is to be delivered:
The first position: It is mustahabb to offer two sermons. This is the position of the Shafi`i school and the majority of the Salaf.
The second position: One sermon is what is legislated. This is one position among Hanbalis and the position of choice of Ibn `Uthaymin.
Thirdly: Acts Legislated During an Eclipse
During an eclipse, it is mustahabb to make mention of Allah the Exalted, supplicate, ask for forgiveness, give charity, and to draw closer to Allah through every means. This is explicitly mentioned by Hanafis, Malikis, Shafi`is, and Hanbalis.