No. There is no language as a Kazakhstan Language. However, there is a Kazakh language, which is the state language of Kazakhstan.
Kazakhstan is actually a multilingual country, as it's home to over 100 different ethnic groups. How do they communicate? Which language do the Kazakhstanis speak?
As you know Kazakhstan has been part of the USSR, whose state language was Russian. Therefore many people still speak it all over the former Soviet Union.
That's why the Russian language kept its status as the official language in Kazakhstan even after its independence, which makes Kazakhstan a bilingual country.
Now people of other ethnic groups other than Kazakhs or Russians, speak one or both of the official languages along with their own mother tongue.
I often read here and there people asking if Kazakh and Russian languages are similar in any way. The answer is no. Kazakh language belongs to Altaic languages, while Russian is an Indo-European. Kazakh is a Turkic language group, while Russian is a Slavic.
Saying that many major ethnic groups in Kazakhstan belong to the same Altaic-Turkic group of languages, so if you speak any Turkic language and try hard enough, you too can understand many people's speech while in Kazakhstan.
Don't worry and don't let it put you off from visiting Kazakhstan, if you don't speak Kazakh or Russian. Many young people speak at least one sometimes a few world languages.
To start with if you speak English you will get by. If you are an experienced traveler and you know some sign language, you can mix all that knowledge during you communication process.
If you let yourself have a shot of vodka to boost your confidence ;), you can get over any language problems, as people are very friendly and they try to understand and help you out.
Here is a video with non-kazakhs, who have been learning the language while living in Kazakhstan. Anyway listen to how it sounds.
By the way it's a 7 minute video, so be aware just in case if you have to rush out.