This question takes its classic form by moving the phrase in question to the end of the sentence.
- These are the problems with which we should deal.
- No, these are the problems that we should deal with.
- No, no, no, these are the problems we need to deal with.
Many people will tell you that it is not permissible to end a sentence with a preposition. However, as Winston Churchill reportedly once said:
This is a rule up with which I will not put!
Most authorities have dispensed with any notion of absolutism on this issue, and I think it is a good idea for everyone else to do so as well.
Nevertheless, the better style is always that which is best match the expectations of the audience. Here, it is not possible for me to agree that using the "with which" formulation (which is the basic solution to the ending-preposition problem) is necessarily "better."
But it is, arguably, a "higher" form of style and in contexts where formalism is privileged (slightly) over directness of speech, this option is the less risky.
For contexts in which directness of speech may be valued at the (occasional) expense of older notions of formalism, whether to use a "that" or not is up for grabs. Many people say that explicitly using all of the "that's" that would otherwise simply be implied is a good idea for second-language readers. Many people are also of a split mind about it.
Newspapers and other publications that are normally tight on page space would prefer that their writers habitually lean toward economy of words and, thus, leave all "unnecessary" "that's" out of their copy. Textbook writers and other technical writers tend to favor erring on "the safe side" and including them.
Regionally and within certain other academic or publishing traditions, there may very well be stronger preferences than what I am representing here. Hopefully, some others will share their stories, opinions, impressions and so forth.