Hello.

I'll give you some guidelines how to verify everything is OK.

You want to check postfix logs and mail headers.
Make sure you have setup SPF-record properly. Beside that it's recommended to ensure that reverse DNS lookup is working for you. DNS lookup works by PTR records, so to check it, make sure your Droplet is named same as your domain. After that check is there record under Networking -> PTR records.

As it works correctly for Gmail, it's very high chance that it's setup correct.

There is a "problem" with Outlook.com. Outlook.com doesn't trust emails that are sent from IP addresses that didn't sent emails previously.
It needs some time to build trust with them and Outlook.com Postmaster should help you with it.
You want to pay high attention to Email I send is being blocked or junked by Outlook.com..
Quote from Are you sending email from new IPs?:

IPs not previously used to send email typically don't have any reputation built up in our systems. As a result, emails from new IPs are more likely to experience deliverability issues. Once the IP has built a reputation for not sending spam, Outlook.com will typically allow for a better email delivery experience.

And little further you can see the notice:

Note: don't forget to update your Junk Email Reporting Program (JMRP) account with the new IPs. To update or set up a JMRP account, click here.

Try following linked guide, it can help y setup mail server to send correctly to Outlook.com.

I've been working on the same issue, and I've contacted both Digital Ocean and Microsoft's support teams. I already had SPF and DKIM set up properly before reaching out and I'd suggest you do the same. You can view one of your server's messages in Gmail, go to the down arrow beside the reply button, and select "Show original" to see if your SPF and DKIM settings are correct. For a more detailed analysis, send an email from your server to check-auth@verifier.port25.com and you'll receive a return email with an evaluation of your settings.

My initial error was 550 SC-001 when sending to outlook.com, which is due to IP reputation. You can look up your error code at http://mail.live.com/mail/troubleshooting.aspx#errors. I thought DO's support was blowing me off when they suggested requesting that MS remove my IP from their blacklist (since DO is the true owner of the IP), but MS actually responded to that request within hours and it did resolve the initial error message. If you're getting error 550 SC-001 or a related error, you can submit that request here:

http://go.microsoft.com/fwlink/?LinkID=614866&clcid

The response from MS will encourage you to join the JMRP and SNDS programs, which I did and hope it will prove useful one day. Don't expect that to help debug further issues, though. Also, past the point of removing your IP from their blacklist, MS support will only send canned responses that refer you to their documentation.

After being removed from their IP blacklist, I got a couple of messages to show up in my Junk folder on outlook.com. Then they stopped being delivered at all despite my mail log showing "queued for delivery" while talking to MS's mail server. What I found just now is that the messages are delivered if I add my sending address to the receiver's contacts on outlook.com. I'm hoping that requirement will be lifted after I build a reputation for good sending practices.

Hope this helps...