The eruption of a volcano affects our weather mostly by the big cloud of dust that it sends high up into the air. At first the dust cloud may darken the earth for miles around the volcano. Then the cloud gradually gets thinner as it is mixed up by large winds.
A very large cloud of dust came from the eruption of the volcano El Chicon in southern Mexico in April 1982. This was big enough to be seen in photos taken by our satellites. The cloud rose to a height of about 15 miles. There, a wind stream carried it in a narrow path all the way around the earth. Three weeks later, parts of the cloud could still be seen in photos.
So far I have not seen any report about special effects of that dust cloud on our weather. Some effects are certain to have happened somewhere, just because a dust cloud reflects sunlight and will cause at least a small, local cooling effect. But weather is so variable anyway that some small effect is often difficult to see.