How LL Grammars are subset of DCFL?
You don't have to prove that LL-grammars form a subset of DCFG. It is just a definition. For example according to the Wikipedia:


  Restricting the DPDA to a single state reduces the class of languages accepted to the LL(1) languages. 


This is more about LL-parsers.

LL(1)-grammar alternatively can also be defined as 


  A grammar whose parsing table has no multiply-defined entries is said to be LL(1). The first "L" in LL(1) stands for scanning the input from left to right, the second "L" for producing a leftmost derivation, and the "1" for using one input symbol of lookahead at each step to make parsing action decisions. 


The latter definition is taken from Aho, Sethi, Ullman's "Compilers" book.

In general $LL(k)$ parser may lookahead $k$ tokens to decide what production rule to apply.

However, you might be given a grammar $G$ and prove that this grammar is, for example, LL(1). In this case you could create a parse table and check if it has no multiply-defined entries.