Vinyl records are made of polyvinyl chloride (PVC), which is typically a fairly rigid polymer that is hard to work with. Traditionally, the way around this is to add materials, called plasticizers, that soften the polymer, and make it easier to work with and to mold. Unfortunately, these plasticizers tend to leach out of the plastic over time, which means that it becomes more brittle. Adding more plasticizer to the polymer as it is melting will make it easier to deal with, but you should be aware that there are health concerns related to these materials, which are typically phthalates. This is also one of the reasons why PVC is not recycled often.
Without resorting to this, you may have more luck if you keep the temperature as low as possible: this will minimize the loss of plasticizer as well as the decomposition of the polymer that may occur at higher temperatures. PVC melts at a relatively low temperature (80 °C), so you would have the best luck by placing it in a bowl in a bath of boiling water (the same way people melt chocolate to prevent it from burning). Please be aware that burning PVC can lead to the emission of toxic gases, so you should always maintain the temperature as low as possible and work in a well ventilated area. Personally, I would look for a different polymer to work with.