Hello, I wondered what is the right spelling for the word "heart".
I often see it spelled as "croí" or "chroí", with an "h".

Are both spellings correct?

When standing alone, the word is always "croí". When you see it with the added "h", it is due to its use in a particular phrase, where in some cases there is a sound change in the initial consonant. This occurs, for example, in the traditional endearment "A stór mo chroí" ("O pulse of my heart" = "O my beloved"). This type of change is referred to as "séimhiú" (which comes from the word for "slender" or "mellow"), and occurs quite often in Irish, with a number of consonants in various contexts.

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.

There are quite a few situations where you get lenition/séimhiú at the beginning of words in Irish (as well as in the other Celtic languages). I suggest that you have a look at the on-line grammar at http://www.nualeargais.ie/gnag/gram.htm and click on Initial Mutations (Chapter 14) - Lenition is the first on the list. You can also look up croí on the Teanglann website in the Irish-English dictionary - if you click on the grammar section, you will see the different forms it takes in the nominative and genitive. http://www.teanglann.ie