The names of metric units are usually spelled out.
There are no abbreviations in metric, only symbols. These symbols take periods only at the end of a sentence, never take an s in the plural and are separated from a number by a space. In adjectival uses, the space is replaced by a hyphen.
C (for Celsius) can be used on first reference. Other common symbols such as km/h and mm may be used on second reference when preceded by a number: at 70 km/h, a 105-mm cannon.