This is not odd at all and very common. One easy way to do that is to use an vpn solution, e.g by deploying an openvpn VM from an openvpn AMI (almost everything is configured by default), or by installing another vpn solution on one of your instance. Then, you will only have to log to this vpn and you will be able to resolve your instances with their private IP addresses

However, if you do want your instances to be routed between your LAN/VPC, this is slightly more complicated. you may have to consider using a more complex architecture, still based on VPN but with "Virtual Private Gateway / customer gateway" configuration

The following article has a full explanation on the process involved Technet However the following details the basic steps.

Download Azure Management Certificate

Get-AzurePublishSettingsFile

Import Azure Management Certificate

Import-AzurePublishSettingsFile "${env:USERPROFILE}\Documents\Azure.publishsettings"

Select Azure Subscription

$subscriptionName = (Get-AzureSubscription).SubscriptionName | Out-GridView -Title "Select Azure Subscription" -PassThru

Select-AzureSubscription -SubscriptionName $subscriptionName

Get Azure subscription information

$subscription = Get-AzureSubscription $subscriptionName -ExtendedDetails

$certificate = $subscription.Certificate

$subscriptionId = $subscription.SubscriptionId

Select Azure VNet for which to manage VPN certificates

$azureVNet = (Get-AzureVNetSite).Name | Out-GridView -Title "Select Azure VNet" -PassThru

Import saved copy of user's VPN client certificate

$certPassword = Read-Host "Enter VPN client certificate password" -AsSecureString

Import-PfxCertificate -FilePath "${env:USERPROFILE}\Documents\Azure\P2S VPN\VPN01Client.pfx" -CertStoreLocation Cert:CurrentUser\My -Exportable -Password $certPassword

Select VPN client certificate to Revoke

$vpnCertThumbprint = (Get-ChildItem Cert:\CurrentUser\My | Out-GridView -Title "Select VPN certificate to revoke" -PassThru).Thumbprint

Build web request header

$requestHeader = @{"x-ms-version" = "2012-03-01"}

Revoke a VPN Client Certificate

$revokeVPNCertUri = "https://management.core.windows.net/$subscriptionId/services/networking/$azureVNet/gateway/clientcertificates/$vpnCertThumbprint"

$response = Invoke-RestMethod -Uri $revokeVPNCertUri -Certificate $certificate -Method Post -Headers $requestHeader

Confirm Revoked VPN Client Certificates

$listRevokedVPNCertUri = "https://management.core.windows.net/$subscriptionId/services/networking/$azureVNet/gateway/clientcertificates"

$response = Invoke-RestMethod -Uri $listRevokedVPNCertUri -Certificate $certificate -Method Get -Headers $requestHeader

$response.ClientCertificates.Thumbprint