先把长句化简
Because...,they relied on ... to justify sth. to be doing sth.
1. because从句已清楚；
2. rely on ... to do:
rely on 后面是复合宾语,即宾语（precedents）+不定式充当的宾语补足语（to justify their claim ）,意思是：依赖先例去证明他们的主张.
3. justify ... to be ...
语法结构同上,justify后同样是复合宾语,意思是：证明他们主张是法律；
4. to be merely declaring pre-existing law and not making up their own law是并列的两个不定式补足语,作their claim的补足语：
a）to be merely declaring pre-existing law 是仅只宣布着的先前的法律.
b）not (to be) making up their own law是没有修订着的他们自己的法律.
c）to be claring ... and not making是进行时态,即：正公布的...而不是正修订的...
结构清楚,翻译就简单多了：
因为国王的法官们缺乏英国法律的书面记录,他们就依赖先例[过去的判决]去证明他们的主张仅只是正宣布着的先前的法律而不是正修订着的他们自己的法律.