In a lot of business code reviews are compulsory (or at least some sort of inspection) and often required for accreditation.
There is often a convenient change set from version control tools such as SVN/GIT etc that can be used to define review content and these change set often have associated diff tools to help.
On previous projects we'd just branch every change in SVN and get someone else to check the changeset to merge onto trunk.
Saying that we are currently using JIRA with associated code review tools and this is great and provides a nice log of comments and responses which we can present to QA (as review evidence) on a simple web interface.
Reviewing the code is a great way of sharing the knowledge in addition to finding bugs earlier (if done well).
We do line-by-line code reviews of all changes (high-assurance software, so we have to - but it's worth it anyway.) Some kind of tool is useful, since it allows you to (a) review code on your own schedule, and (b) stores the comments (for later review, if someone is on holiday, so that you can refer back to your initial comment when reviewing an improvement, ...)
That said, code reviewing is a skill (and perhaps a talent), and not the same skill as software development; a degree of training - if just internal and informal - is necessary.
Thorough code reviews take a significant fraction of the effort needed to write the code in the first place; direct tooling costs are going to be insignificant in comparison. (And note that e.g. Gerrit and even GitHub both offer some degree of free tooling.)
We use Crucible for our code reviews and I have been happy with it. We use the Atlassian 'stack' so it integrates well with Jira and Bitbucket, which is nice.
It is nice to have all changes highlighted and the ability to make comments on line levels or file level so that the author, and team, can see what is going on and get good feedback.
We do code reviews, but I wouldn't use an automated tool for them. We treat code review as an opportunity for communication, learning, institutional knowledge transfer, and architecture review -- not as a way to enforce code style or catch memory leaks. That's what linters are for.
I pay for code reviews in the UK from a company called Atlas Computer Systems Ltd https://www.atlascode.com/
They do code reviews for startups to make sure your code is high quality and isn't building technical debt.