The best time to climb Elbrus on foot is July and August. The snow level is minimal, weather is most stable. June, September and first half of October are also rather favorable. It is colder and the weather may be changing, but there is not too much snow on the slopes and still many chances to reach the summit.
Attempting it in spring is rather challenging, especially from the north side. The road is blocked by snow and participants have to walk the last 13 km to the base camp carrying their backpacks themselves. This is rather exhausting taking into account the altitude difference of 550 m over that distance. Temperature on Elbrus in spring can fall as low as -40. Add storm winds and icy slopes which can be really dangerous for novices and require fixing a rope.
We don't recommend climbing Mt. Elbrus in winter. There are very few chances to reach the summit because of extremely low temperatures, snow storms and blue ice on slopes.