We need to clear up a couple of points to begin with. In the standard way in which logic is treated today, we distinguish between the object language and the metalanguage. This amounts to distinguishing between saying something in a language and saying something about a language. Material implication is a connective within the object language of classical logic. What you are calling logical implication, which is also sometimes called logical consequence, is a statement about the propositions in a language. Material implication is sometimes written as P ⊃ Q, or as P → Q, or as P ⇒ Q.
Another important distinction is between syntax and semantics. Syntactically, we say that some set of propositions Γ proves a proposition β, or β is derivable from Γ, or β is a theorem on Γ, if there is some way to prove β from Γ using the axioms or rules of our logic. This relation is written as Γ ⊢ β. Semantically, we say that β is the logical consequence of Γ, or the semantic consequence of Γ, or that Γ entails β, if the truth of Γ guarantees the truth of β according to some semantic framework. This relation is written as Γ ⊨ β. The usual framework is called model theory and we say that an argument is semantically valid if every model of Γ is a model of β. First order classical logic is sound and complete, which means that Γ ⊢ β and Γ ⊨ β will always both be true or both be false.
Material implication, like other connectives, can be defined either syntactically or semantically, and the two are provably equivalent. For comparison, conjunction can be defined syntactically using an introduction rule: P; Q therefore P ^ Q and an elimination rule: P ^ Q therefore P; Q. These rules are sufficient to characterise the classical conjunction connective exactly. Conjunction can be defined semantically by a truth table: T/F/F/F. The two can be proved to be equivalent using techniques that you will typically find in an intermediate level textbook. In the case of material implication, it can be defined syntactically by the introduction rule Γ ⋃ {P} ⊢ Q therefore Γ ⊢ P → Q, and the elimination rule P; P → Q therefore Q. Semantically, its truth table is T/F/T/T. The introduction rule looks a little odd, but it is provable using the deduction theorem.
Now that we have this in place, it is straightforward to answer your questions. Material implication has a truth table because it is a truth functional connective in the object language. The derivability and logical consequence relations do not themselves have truth tables, because they are relations between sets of propositions. To understand the relationship between them, take a look at what happens to the introduction rule for material implication when Γ is empty. It simplifies down to P ⊢ Q therefore ⊢ P → Q. In other words, material implication is provable in just those circumstances in which Q is provable from P. And since by soundness and completeness we can substitute ⊨ for ⊢, it also follows that P ⊨ Q therefore ⊨ P → Q. In other words, P → Q is a tautology in just those circumstances in which Q is the logical consequence of P.
So, as an example, suppose we wish to test whether this argument is valid:
P1: a → b
P2: c → d
Therefore
C: (b → c) → (a → d)
First we form the corresponding conditional of the argument. This is done by conjoining the premises and having these materially imply the conclusion, thus:
((a → b) ^ (c → d)) → ((b → c) → (a → d))
Then we plug this into a truth table generator, of which there are many on the Web. This yields the following. As you can see, the corresponding conditional has T in all rows, so it is a tautology, and the argument is valid.
+---+---+---+---+---------------------------------------------+
| a | b | c | d | (((a → b) ∧ (c → d)) → ((b → c) → (a → d))) |
+---+---+---+---+---------------------------------------------+
| F | F | F | F | T |
+---+---+---+---+---------------------------------------------+
| F | F | F | T | T |
+---+---+---+---+---------------------------------------------+
| F | F | T | F | T |
+---+---+---+---+---------------------------------------------+
| F | F | T | T | T |
+---+---+---+---+---------------------------------------------+
| F | T | F | F | T |
+---+---+---+---+---------------------------------------------+
| F | T | F | T | T |
+---+---+---+---+---------------------------------------------+
| F | T | T | F | T |
+---+---+---+---+---------------------------------------------+
| F | T | T | T | T |
+---+---+---+---+---------------------------------------------+
| T | F | F | F | T |
+---+---+---+---+---------------------------------------------+
| T | F | F | T | T |
+---+---+---+---+---------------------------------------------+
| T | F | T | F | T |
+---+---+---+---+---------------------------------------------+
| T | F | T | T | T |
+---+---+---+---+---------------------------------------------+
| T | T | F | F | T |
+---+---+---+---+---------------------------------------------+
| T | T | F | T | T |
+---+---+---+---+---------------------------------------------+
| T | T | T | F | T |
+---+---+---+---+---------------------------------------------+
| T | T | T | T | T |
+---+---+---+---+---------------------------------------------+
