You can cycle the Camino from late spring until late autumn, in the other words from mid-June to the beginning of November. Summer tends to be very hot and the trail gets very busy during the holiday. Temperatures in July and August reach 35-36 degrees. I did the Camino once in the high season and yes it was challenging because of the strong sun and heat. Most of the route, especially for cyclists goes through exposed spaces; so a better time to set off on the journey for you is spring or autumn. June, September, and October are still quite sunny and warm with temperatures sometimes reaching 26 degrees. From mid-October until the very beginning of November the temperature usually drops, it might be really cold in the morning (around 8 degrees), but sunny and warm during the day. At any time of the year, even if you are wearing strong sun protection cream, you will come home with some suntan.
I have cycled in the spring and in autumn and in my opinion June or September until mid-October is the best. The end of October and very beginning of November is brilliant as well but much colder. When choosing the right time to set off take everything into account, not only the weather but also the number of people on the route and accommodation possibilities. There are all season albergues, but many close at the beginning of October, so bear this in mind. There are also fewer pilgrims on the way in late spring and late autumn.
I really don’t recommend cycling the Camino at any other time of the year, because to reach Santiago you have to cross three mountain ranges – The Pyrenees, The Montes de Leon, and Sierra de Ancares/del Caurel. The elevation of the terrain is very high, so even if climbing is possible, sliding down from Roncesvalles Pass, Cruz de Ferro or O’Cebreiro will not be if there is snow or if it is raining heavily.
Before you set off on the journey which is not only a bit risky in early June (you might be surprised by snow lying in the higher mountain ranges or a sudden change in the weather) and the end of October but actually at any other time of the year as well: ALWAYS ask about the weather conditions at the Pilgrim’s Office in Saint-Jean-Pied-du-Port. The atmospheric conditions in the Pyrenees might change suddenly leaving you helpless on the mountainside. There are fatal accidents on the Roncesvalles Pass, so consult with the office about the route and take their advice. The Pyrenees is a very beautiful and impressive mountain range that I adore but with a change in the weather, they may become a deadly trap.
In Navarra and La Rioja you could experience a mixture of sun and rain depending on personal luck and amount of sins, in the Burgos area it is usually raining (personal experience; statistics don’t prove it), in the flat as a pancake and woodless region of meseta (Palencia, Castilla y Leon) more often than not the sun blazes, although it is even more interesting when the wind blows.
Between Astorga and Ponferrada you have to cross the high mountain range of Montes de Leon, wild and impressive. Before you start to climb to Cruz de Ferro ALWAYS ask about the weather in Rabanal del Camino.
Soon you will have to cross your final mountain range between Vega de Valcarce and Tricastella, so check if the weather is good before setting off.
Galicia, oh Galicia is green and rainy and against its nature has always treated me kindly with the sun, until 2014 when I tried a different route. I have finally gained experience of torrential rain and mist a whole day long with a never-ending eucalyptus forest on both sides. Be warned.