How do I programmatically set the connection string for Entity-Framework Code-First?

you should use the EntityConnectionStringBuilder class

string providerName = "System.Data.SqlClient";
string serverName = ".";
string databaseName = "AdventureWorks";

// Initialize the connection string builder for the
// underlying provider.
SqlConnectionStringBuilder sqlBuilder =
new SqlConnectionStringBuilder();

// Set the properties for the data source.
sqlBuilder.DataSource = serverName;
sqlBuilder.InitialCatalog = databaseName;
sqlBuilder.IntegratedSecurity = true;

// Build the SqlConnection connection string.
string providerString = sqlBuilder.ToString();

// Initialize the EntityConnectionStringBuilder.
EntityConnectionStringBuilder entityBuilder =
new EntityConnectionStringBuilder();

//Set the provider name.
entityBuilder.Provider = providerName;

// Set the provider-specific connection string.
entityBuilder.ProviderConnectionString = providerString;

// Set the Metadata location.
entityBuilder.Metadata = @"res://*/AdventureWorksModel.csdl|
                        res://*/AdventureWorksModel.ssdl|
                        res://*/AdventureWorksModel.msl";
Console.WriteLine(entityBuilder.ToString());

using (EntityConnection conn =
new EntityConnection(entityBuilder.ToString()))
{
conn.Open();
Console.WriteLine("Just testing the connection.");
conn.Close();
}

In Entity-Framework Code-First use SqlConnection. You cannot use EntityConnectionStringBuilder because in code-first there is no metadata files.

You can also set it directly on your context. You have to attach the mdf to the SqlServer instance you want to use first. Not exactly elegant but it worked for me

public void DoImportWork()
{
   var ctx = new StatisticsContext(); << your DbContext 

   ctx.Database.Connection.ConnectionString = @"Data Source=localhost\SQLEXP;AttachDbFilename=""C:\Program Files\Microsoft SQL Server\MSSQL11.SQLEXP\MSSQL\DATA\StatisticsData.mdf"";Integrated Security=True";

   ctx.Database.Connection.Open();
}

as usual EF will auto-generate everything when you add your first row to the context.