If national character is the way others outside of our country perceive us, then no, climate of the country does not influence national character. The behaviors of individuals and groups in response to a national climate might play into this, but it doesn not predict or define it.
If national character is defined as the common personality characteristics of an entire society or country it is unlikely that any country but the smallest could possibly have a common personality. However, in countries in which one part is on the sea, for instance, the inhabitants of this area who are fishermen are certainly different from the ones who are farmers, for instance. Their cultural differences are evident in their speech, attitudes, and pleasures.
I do not think that the climate of a specific country influences its national character. It certainly has an influence on how they do things but that's about it. It is not related to how other countries perceive national character of other countries.
In my opinion, it does not. I think that climate only gives people outside the country a reason to stereotype the people of the country. In other words, for example, people in the West have had a stereotype that says that people from the tropics are lazy because their climate is hot and they don't like to work hard in the heat. I do not think that the heat has really made these people lazy. I think that they appeared lazy to Westerners because they were not industrialized people with the habits of working by the clock, etc.