1. The service provides several APIs through its contract, which we can use to verify or relay Bitcoin transactions. These APIs can be used in your DApp to achieve Bitcoin-related verification. Let's understand the important functions exposed by the BTCRelay contract through an example that uses them.
2. Use the verifyTx function to verify a Bitcoin transaction. A transaction is considered as verified only if it has at least six confirmations. The function returns the hash of the verified bitcoin transaction. It returns 0 if the verification fails:

verifyTx(
    rawTransaction, // Raw transaction bytes - bytes
    transactionIndex, // Index of the transaction in Block - int256
    merkleSibling, // Merkle proof sibiling hashes - int256[]
    blockHash // hash of the transaction block - int256
) returns (uint256)

3. To verify and relay the transaction to a specific contract, use the relayTx function provided by BTCRelay. The function is very similar to verifyTx but accepts one more parameter as contractAddress. The target address should have a function called processTransaction. The relayTx function returns either the result of processTransaction or an error code:

relayTx(
 rawTransaction, // Raw transaction bytes - bytes
 transactionIndex, // Index of the transaction in Block - int256
 merkleSibling, // Merkle proof sibiling hashes - int256[]
 blockHash, // hash of the transaction block - int256
 contractAddress // Address of the target contract
) returns (int256)

// Target contract function
// Will be invoked by relayTx
processTransaction(
 bytes rawTransaction, 
 uint256 transactionHash
) returns (int256)

4. Use the following functions provided by the service to set and get the block header of a block hash:

// Get header
getBlockHeader(
    blockHash // Hash of the block
) returns (bytes)

// Set header
storeBlockHeader(
    blockHeader // raw block header - bytes
) returns (int256)

// Set multiple headers at once
bulkStoreHeader(
    bytesOfHeaders, // raw block headers one after another - bytes
    numberOfHeaders // number of headers
) returns (int256)

5. getBlockHeader returns the header as bytes if it is found. It returns a 0 result with a length of 80 bytes if the header does not exist. The result is 0 if the payment is insufficient. The payment can be calculated by calling the getFeeAmount function:

getFeeAmount(blockHash) returns (int256)

6. There are other functions that are used to read and write more data about bitcoin transactions:

// Returns the hash of given block height
getBlockHash(blockHeight) returns (int256)

// Returns the difference between the chainwork
// of latest and 10th prior block
getAverageChainWork()

// Returns the hash of latest block
getBlockchainHead() returns (int256)

// Returns the block height of latest block
getLastBlockHeight() returns (int256)

7. BTCRelay also includes functions to help relayers with incentives. Relayers are those who update the contract with recent blockchain transactions:

// Store single block header and store its fee
storeBlockWithFee(blockHeader, fee) returns (int256)

// Sets fee and the recipient for the given block hash
changeFeeRecipient(blockHash, fee, recipient) returns (int256)

// Get the fee recipient for the given block hash
getFeeRecipient(blockHash) returns (int256)

// Get the amount of fee for changeFeeRecipient
getChangeRecipientFee() returns (int256)

8. The service can be used in your application by specifying its deployed address and ABI. Consider the following example as a template for calling the functions:

var btcRelayAddr = "0x...";
var btcRelayAbi = [...<BTCRelay_ABI>...];

var btcRelay = web3.eth.contract(btcRelayAbi).at(btcRelayAddr);

btcRelay.verifyTx.call(
    transactionBytes,
    transactionIndex,
    merkleSibling,
    transactionBlockHash,
    { from: '0x..', ...}
);

9. Refer to the official GitHub repo (https://github.com/ethereum/btcrelay) for more information related to Ethereum contract and internals.