I'm not sure if it's possible to reach the top of the spire by only fighting, but I can tell you what I did to reach the top of the spire.
1) I joined a FS which got a bronze medal weekly.
2) I played the spire, but conceived. Getting to the top of the 1st level wasn't too difficult. In fact getting to the top of the 2nd level wasn't too difficult for me. I soon achieved that, but with conceiving
3) I crafted Fire Phoenix. And used the diamonds I won in the spire to craft Phoenix artifacts.
4) Once I was able to get to the top of the 2nd level I joined a FS which got a silver medal every week. And a blueprint. The blueprints I also used for crafting Phoenix artifacts.
5) Once my Fire Phoenix was at level 10 (six months after I started playing the game) I started fighting in the spire. I also used military booster buildings for that (normally I use 2 UUU's (or 1 DA), 1 ELR and 1 MMM). And started fighting in the spire and doing the 3rd level. However there are still situations where I prefer to conceive instead of fight.
6) Once I was able to get to the top of the 3rd level I joined a FS which got the gold medal weekly.