most consulting firms focus on three CV areas:
- Education: Top-tier universities, grades above average (doesn't need to be top of class if you fulfill the requirements of the other areas) and no obvious weak spots (especially maths)
- Work experience: If you apply for a full-time position, you need relevant work experience (consulting, strategy, project management...) at well-known companies (less relevant if you graduated top of class)
- Other: Extra-curricular work, leadership experience (sports etc.) and personal interests - HR usually looks for well-rounded personalities. However, this is the least important area, so if you impress in areas 1 and 2, you needn't worry too much about this one
Given your (short) description: I don't see a problem with education, however, given you only completed two internships with not so well-known companies, I would say it is going to be a bit more difficult for you to pass the CV screening.
What I would advice you to do: Apply at some tier-two consulting firms and see if you get invited. If not, you need to strengthen area 2, meaning you should do one or two internships with well-known companies, then apply again (you don't get blacklisted, unless you get invited for an interview and do not pass it).
Hope this helps.