You can use the explode function to split the string into an array based on a delimiter in you particular case, the apostrophe would work to delimit the string so that the following code would yield your answer:
$tokens= explode("'", "Hello my name is 'Kate' and im fine");
//The value you require is now found in $tokens[1];
echo $tockens[1];
Alternatively you could use the preg_match to store regular expression matches against particular groups in the regular expresion:
$pattern = "Hello my name is '(.*)' and im fine";
preg_match ($pattern , "Hello my name is 'Kate' and im fine", $matches)
//The value you require is now found in $matches[1];
echo $matches[1];