It depends on what do you mean by "construct". There is a paper of Chevalley "Certain schemas de groupes semi-simples", where he gives a uniform construction, but it is not something you can easily put into your computer ;). Much more computational approach is in Takeuchi's "Generators and relations for hyperalgebras of reductive groups", where he construct a Hopf algebra which is dual (in some certain sense) to the coordinate Hopf algebra of a reductive group.

It depends on what do you mean by "construct". There is a paper of Chevalley "Certain schemas de groupes semi-simples", where he gives a uniform construction, but it is not something you can easily put into your computer ;). Much more computational approach is in Takeuchi's "Generators and relations for hyperalgebras of reductive groups", where he constructs a Hopf algebra which is dual (in some certain sense) to the coordinate Hopf algebra of a reductive group.