The relationship between America and Russia does look like it is going to change with Trump being a big fan of Putin. Which is very worrying for the people of America. The last thing they need is more worry thanks to the Secretary of State having ties directly with Putin. For this reason Trump shouldn't hire him, but we know how Trump wants to go about things so it probably means he will hire him.
No. Considering his ties to Russia, Rex Tillerson should not be confirmed as Donald Trump's secretary of state. The United States must maintain its full sovereignty and make sure that all of its elected and appointed officials have only the best interests of the country in mind. Conflicts of interest like this should be avoided.
Many of Trump's nominees are going to have ties to other countries. Ties like that would in some ways make him qualified to be the Secretary of State. However, there are other options that would be better. Mitt Romney is pretty popular, both domestically and abroad. He should be the choice over Tillerson.
As Secretary of State, you are the face of the United States to foreign entities. You have to be objective and trusted to deal with the best interest of the United States. Because of Rex Tillerson's ties to Russia, this objectivity is in question and there is a lack of trust in his judgement. He should not be Trump's Secretary of State