I got 90% in two of my second year BSc essay exams, 80% in my favourite (Forensic Psych) BSc (third year) essay, and in my MSc got 78% for my first (Sport Psych) essay, 80% for my two Research Methods essays, and 85% for my Exercise & Health essay. So it is possible.
How to do it?
Well...I didn't ever go to lectures at undergrad or to the library (3x in 3 years...do as I say, not as I do). But what I did do was use readings at home. Primary sources as much as possible - uni library/online journal sources obviously, Google Scholar, etc etc. Try to avoid relying too much on secondary sources - textbooks etc - but they are useful for basic overviews, and for giving you ideas for what to look up or focus on.
At MSc level I did attend every lecture/seminar and went to library every time I was in uni (only 2 days a week for full-time). But pretty much the same principle - primary sources, online journals.
For higher marks, I recommend:
A good introduction! I am lazy with introductions. But keep working on them and rewriting. How they are supposed to be will differ. Personally I hate the 'This essay will discuss' approach, but some expect that.
Good critical analysis skills - link your points together. Make your arguments flow. Discuss counterarguments, but back up your points for why you believe your argument is stronger. Discuss where relevant: methodology concerns, limitations of research, generalisability, practical implications of arguments, etc, what your argument would mean for subject.
A good conclusion! Again, see above re. introduction. But a good conclusion should sum up your argument effectively.
As an example of my highest MSc essay (85%) (exercise & anorexia), it went basically along these lines:
Introduction - Define general topic (positive correlations with exercise in healthy population). Link to EDs (less positive associations). Define anorexia nervosa; prevalence, exercise stats, treatment etc. State question: Exercise suggested as useful addition to treatment of AN, raising question: how can a symptom be used as a treatment?
Point 1 - Furhter detail on exercise in individuals with AN; stats & correlations. Explain why therefore exercise restriction often recommended.
Point 2 - Introduce more recent notion of incorporating exercise into treatment programmes. Explain debate over issue, using referenced arguments (but include limitations, e.g. non-clinical samples --> not generalisable, may differ).
Point 3 - Discuss studies which haven't found negative correlations between EDs/exercise. (Again use counter-arguments).
Point 4 - Explain conceptual framework for essay topic (e.g. here, benefits of exercise in treatment through enhancing self-esteem, body image, mood)
Point 5 - Counterargument - discuss other explanations of findings (e.g. ED/exercise --> improved mood. But may perpetuate dysfunctional ED/exercise-related thoughts? Reinforcing effect of exercise?) Discuss how reasons for exercising importance as well as amount of exercise, per se (again with references). Explain implications of findings: e.g. treatment programs should aim to help develop more positive exercise beliefs
Point 6 - Flow from mention of treatment programs to discussing how to increase effectiveness of exercise interventions in ED treatment programs - e.g. what studies have included and findings of studies. Again consider caution of studies. Sum up why psychoeducational component in treatment is ideal setting to learn to incorporate exercise in adaptive manner.
Point 7 - Future research recommendations E.g. base interventions on theory and guidelines, standardised treatments, consider study eligibility and who would benefit from particular intervention.
Point 8 - Go further into methodological limitations (with examples/references), and again recommendations for future research.
Point 9 - Despite limitations, some evidence has supported... etc etc. Summarise positive findings from intervention studies, and why this may be case, although considering counter-arguments.
Point 10 - Further (linked) concern (in my essay this was paragraph on another ethical concern), followed by counterargument and references, and conclusion that there is clear rationale for (my argument).
Point 11 - Point on how to limit above concerns (e.g. here, by considering type/amount of exercise), considering stats/findings from studies using various types, and why one may be more effective than other. Again further research recommendation.
Conclusion - Sum up - e.g. contentious issue; as yet, no consensus; ethical concerns; but potential benefits of incorporating exercise for ED. Future research recommendations
Obviously that is pretty long and is based on my own essay so will vary! But basically, lots and lots of referencing, evidence for points and implications of what findings mean, counter-arguments, counter-counter arguments, theoretical rationales and how they work in practical, discuss study limitations and make recommendations etc.