Convert underscore to PascalCase, ie UpperCamelCase

$ echo "this_is_the_string" | sed -r 's/(^|_)([a-z])/\U\2/g'            
ThisIsTheString

Substitute pattern
(^|_) at the start of the string or after an underscore - first group
([a-z]) single lower case letter - second group
by
\U\2 uppercasing second group
g globally.

Since you're using bash, if you stored your string in a variable you could also do it shell-only:

uscore="this_is_the_string_to_be_converted"
arr=(${uscore//_/ })
printf %s "${arr[@]^}"
ThisIsTheStringToBeConverted

${uscore//_/ } replaces all _ with space, (....) splits the string into an array, ${arr[@]^} converts the first letter of each element to upper case and then printf %s .. prints all elements one after another.
You can store the camel-cased string into another variable:

printf -v ccase %s "${arr[@]^}"

and use/reuse it later, e.g.:

printf %s\\n $ccase
ThisIsTheStringToBeConverted

Or, with zsh:

uscore="this_is_the_string_to_be_converted"
arr=(${(s:_:)uscore})
printf %s "${(C)arr}"
ThisIsTheStringToBeConverted

(${(s:_:)uscore}) splits the string on _ into an array, (C) capitalizes the first letter of each element and printf %s ... prints all elements one after another..
To store it in another variable you could use (j::) to joins the elements:

ccase=${(j::)${(C)arr}}

and use/reuse it later:

printf %s\\n $ccase
ThisIsTheStringToBeConverted

Here's a Perl way:

$ echo "this_is_the_string" | perl -pe 's/(^|_)./uc($&)/ge;s/_//g'
ThisIsTheString

It can deal with strings of arbitrary length:

$ echo "here_is_another_larger_string_with_more_parts" | 
    perl -pe 's/(^|_)./uc($&)/ge;s/_//g'
HereIsAnotherLargerStringWithMoreParts

It will match any character (.) that comes after either the start of the string or an underscore ((^|_)) and replace it with the upper case version of itself (uc($&)). The $& is a special variable that contains whatever was just matched. The e at the end of s///ge allows the use of expressions (the uc() function in this case) within the substitution and the g makes it replace all occurrences in the line. The second substitution removes the underscores.