Q:

What does てよりかか mean?  Found in the Shinmeikai dictionary

The Shinmeikai dictinary defines 椅子 as:
腰を掛け（てよりかか）るための家具
What does てよりかか mean there? 

A:

腰を掛ける (here in て-form) means to sit and 寄りかかる means to lean on. 
A chair is a tool where you sit and rest your back against the back of the chair. 

A:

Nothing much to add to @永劫回帰's answer itself, just a follow-up about the format:

腰を掛け（てよりかか）るための家具

It's almost like writing in English:

a piece of furniture for sitt(ing and reclin)ing on

Of course you usually don't want to write such a Frankenstein sentence in English, but Japanese has no spaces between words and can be broken anywhere, so why not?  As a result, we don't hesitate to insert fragments in parentheses wherever both ends meet around them. 

小さ（くて見づら）い字
ブザーが聞こえ（なかっ）た時
無（理のない）課金

