I'm currently reading Rotman's "An Introduction to Homological Algebra" and I'm getting into alot of the category theory concepts and I was wondering if understanding category theory fully before understanding Homological Algebra is a good strategy or is it ok to understand homological algebra while understanding the category theory concepts as you go along.
In other words, my question is whether or not having an understanding of category theory prior to doing homological algebra is the best route.
As a contrast to the comments, category theory has grown immensely since its origin in algebraic topology. I'm very interested in category theory, but for the areas I'm interested in it is not at all clear how homological algebra would be relevant. Conversely, while certainly useful for homological algebra, those areas aren't more useful for homological algebra than any other branch of math. For example, understanding the notion of an adjunction would certainly be useful for homological algebra, but only because it's useful for seemingly every branch of mathematics.
In particular, homological algebra is intensely focused on the concept of an abelian category, but for me, I virtually never care about abelian categories. I often prefer categories that are cartesian closed, but the only cartesian closed abelian category is the trivial abelian category. The (fairly artificial) concept of an AT category (inadvertently?) drives this point home. An AT category is a simultaneous generalization of an abelian category and a pretopos. Indeed, an AT category is an abelian category if and only if $A\times 0\cong A$, and it is a pretopos if and only if $A\times 0\cong 0$ each for every object $A$.1 On the one hand, this shows a large amount of similarity between abelian categories and pretoposes. On the other hand, every AT category is precisely the product of an abelian category and a pretopos. Effectively, abelian categories and pretoposes are disjoint subareas of category theory. It should come as no surprise to you now that many of the kinds of categories I find interesting are pretoposes.
This was a long winded way of saying the following: 1) understanding category theory "fully" is not a very realistic goal, 2) large areas of category theory have little relevance to homological algebra, 3) and, vice versa, homological algebra is not obviously useful for large areas of category theory. You can definitely learn homological algebra without category theory, and you definitely don't need to understand category theory "fully" to apply it to homological algebra. My impression is that categorical notation and concepts have more or less completely overtaken homological algebra and, without a doubt, category theory is very broadly useful beyond homological algebra, so it will be in your long-term interest to become familiar with it. The best approach is likely the one the book is taking: to introduce the basic concepts of category theory in a particular context where you'll have (relatively) concrete examples and some context for why the concepts are useful. My only warning is that one shouldn't confuse the parts of category theory used in such an endeavor with all of category theory. Category theory outside of abelian categories has a very different "feel" than abelian category theory, though there is of course a lot of overlap.
1 Cartesian closure implies $A\times 0\cong 0$ for every $A$ demonstrating my earlier claim.