You need to set up both an MX and TXT record for your domain. On domains.live.com, you need to find the values for these fields in the "Your Domains" section. You should find an MX record that looks like 123456.pamx1.hotmail.com and, further down the page, a TXT record that looks like v=spf1 include:hotmail.com ~all and another that looks like v=msv1 t=12356ca464adfg You need to find the unique values for your domain. You can then add the records where you manage the DNS for your domain.
Are you using DigitalOcean's control panel to handle DNS? If so, take a look here for how to use it:
https://digitalocean.com/community/articles/how-to-set-up-a-host-name-with-digitalocean
