Some of the bad omens in Julius Caesar include the thunderstorm, the seer warning Julius Caesar to beware of the ides of March, an earthquake, and strange behavior of animals.
Julius Caesar by William Shakespeare is not set in Elizabethan England, but in ancient, pagan Rome, and is based on ancient sources such as Plutarch. In this period, most Romans believed in the existence of a wide variety of gods, goddesses and spirits, each of whom might actively intervene in human lives either in a benevolent or malevolent manner. Because the gods were powerful and capricious it was important to try to know their intentions and what they wished of you.
The Romans believed that the gods often communicated with humans by sending various signs or signals that they called omens. The study of such communications was called "divination" and came in many different forms.
First, because the heavens were believed close to the abodes of the gods, astrology was very important and certain astrological signs were considered good or bad omens or signs of the gods' intentions. Also, anything that violated the course of nature, such as a two-headed chicken, animals behaving strangely, major storms, or eclipses could be bad omens. Finally, whenever animals were sacrificed to the gods, an expert haruspex would exam the liver of the animals for good or bad omens.