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The algebraic groups by definition come as algebraic varieties; the homomorphisms there are considered as morphisms of varieties. So they come with some basic tools of algebraic geometry. But I have never gone into serious study of algebraic geometry. I don't know how much of algebraic geometry is needed to study basic theory of algebraic groups.

Can you suggest very elementary book on algebraic groups?

The book by Borel in first glance I found difficult.

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Let me start by saying that algebraic groups are inevitably a topic of algebraic geometry, however you can have a lot of fun with them over $\Bbb C$, in which case they can be studied classically. Even in that case however, I am afraid it will eventually be unavoidable to study some algebraic geometry because otherwise you would quite simply be studying groups, not algebraic ones.

I can recommend three books that all are anchored (mainly) in classical algebraic geometry. Unfortunately, my most blazing recommendation is only available in German language, as far as I know.

  1. If you speak German (that is quite a long shot based on your profile), I very much recommend Geometrische Methoden in der Invariantentheorie by Hanspeter Kraft. It is a very clever and self-contained introduction to algebraic groups over $\Bbb C$. It's quite sad that it is only available in German, because I am quite certain it would best fit your needs.
  2. The close second in this contest is Jim Humphrey's Linear Algebraic Groups. It is a little less concrete than Kraft's book, but still mostly classical and does a crash course on everything it requires from algebraic geometry.
  3. I very much like Lie Algebras and Algebraic Groups by Tauvel & Yu. It is not a small book, and the main advantage is not that it finds clever ways around the algebraic geometry, but instead follows what I like to think of as the "French approach"; they simply include a considerable amount of algebraic geometry in the book. This book is notably somewhat Bourbaquesque in the sense that it attempts to take the most efficient path to proving results, sparing out some of the motivation that you might find in the above two mentions. It depends on your taste whether that is something for you or not.

Remark. There is also this primer by Kraft and Procesi. I enjoyed it very much, but I would frankly not recommend it as the first and only material on the topic of algebraic groups; the treatment is a little too classical for that, at least for my taste. Ultimately however, this is something you should decide for yourself.